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If you are interested in Pythagoras's history and The Pythagorean Philosophy and his amazing discoveries in geometry with a detailed explanation, this collection will be useful for you
Study Set Content:
Pythagoras and the Pythagoreans
21
This diagram is identical to the original figure used in the Euclid’s
proof theorem. The figure was known to Islamic mathematicians as the
Figure of the Bride
.
Sketch of Proof.
Note that triangles
4
ADC
and
4
ADE
are congruent
and hence have equal area. Now slide the vertex
C
of
4
ADC
to
B
.
Slide also the vertex
B
of
4
ADE
to
L
. Each of these transformations
do not change the area. Therefore, by doubling, it follows that the area
of the rectangle
ALM E
is equal to the area of the square upon the side
AB
. Use a similar argument to show that the area of the square upon
the side
BC
equals the area of the rectangle
LCN M
.
This stamp was issued by Greece. It
depicts the Pythagorean theorem.
6.2
The Golden Section
From Kepler we have these words
“Geometry has two great treasures: one is the Theorem
of Pythagoras; the other, the division of a line into extreme
and mean ratio. The first we may compare to a measure of
gold; the second we may name a precious jewel.”
Pythagoras and the Pythagoreans
22
A line AC divided into
extreme and mean ratio
is defined to mean
that it is divided into two parts, AP and PC so that AP:AC=PC:AP,
where AP is the longer part.
A
Q
P
C
Golden Section
AP : AC = PC : AP
Let
AP
=
x
and
AC
=
a
. Then the golden section is
x
a
=
a
−
x
x
,
and this gives the quadratic equation
x
2
+
ax
−
a
2
.
The solution is
x
=
−
1
±
√
5
2
a.
The
golden section
20
is the positive root:
x
=
√
5
−
1
2
∼
.
62
The point
Q
in the diagram above is positioned at a distance from
A
so that
|
AQ
|
=
|
P C
|
. As such the segment
AP
is divided into mean
and extreme ratio by
Q
. Can you prove this? Of course, this idea can
be applied recursively, to successive refinements of the segment all into
such sections.
In the figure to the right
Q
1
, Q
2
, Q
3
, . . .
are selected so
that
|
AQ
1
|
=
|
QP
|
,
|
AQ
2
|
=
|
Q
1
Q
|
,
|
AQ
3
|
=
|
Q
2
Q
1
|
, . . .
respectively.
A
Q
P
C
Golden Section
| AP | : | AC | = | PC | : | AP |
Q
1
Q
3
Q
2
20
...now called the
Golden ratio.
Curiously, this number has recurred throughout the devel-
opment of mathematics. We will see it again and again.
Pythagoras and the Pythagoreans
23
The points
Q
1
, Q
2
, , Q
3
, . . .
divide the segments
AQ, AQ
1
,
|
>
AQ
2
, . . .
into extreme and mean ratio, respectively.
The Pythagorean Pentagram
And this was all connected with the construction of a pentagon. First
we need to construct the golden section. The geometric construction,
the only kind accepted
21
, is illustrated below.
Assume the square ABCE has side length
a
. Bisecting DC at E con-
struct the diagonal AE, and extend the segment ED to EF, so that
EF=AE. Construct the square DFGH. The line AHD is divided into
extrema and mean ratio.
A
B
C
D
E
F
G
H
Golden Section
Verification:
|
AE
|
2
=
|
AD
|
2
+
|
DE
|
2
=
a
2
+ (
a/
2)
2
=
5
4
a
2
.
Thus,
|
DH
|
= (
√
5
2
−
1
2
)
a
=
√
5
−
1
2
a.
The key to the compass and ruler construction of the pentagon is
the construction of the isosceles triangle with angles
36
o
,
72
o
,
and
72
o
.
We begin this construction from the line AC in the figure below.
21
In actual fact, the Greek “
Þ
xation” on geometric methods to the exclusion of algebraic
methods can be attributed to the in
ß
uence of Eudoxus
Pythagoras and the Pythagoreans
24
α
A
B
C
D
E
P
Q
Pentagon
α
β
180 − β + 2α = 180
β = 72
A
P
Q
C
Divide a line AC into the ‘section’ with respect to both endpoints.
So PC:AC=AP:PC; also AQ:AC=QC:AQ. Draw an arc with center
A
and radius
AQ
. Also, draw an arc with center
C
with radius
P C
.
Define
B
to be the intersection of these arcs. This makes the triangles
AQB
and
CBP
congruent. The triangles
BP Q
and
AQB
are similar,
and therefore
P Q
:
QB
=
QP
:
AB
. Thus the angle
6
P BQ
=
6
QABAB
=
AQ
.
Define
α
:=
6
PAB and
β
:=
6
QPB. Then
180
o
−
β
−
2
α
= 180
o
.
This implies
α
=
1
2
β
, and hence
(2 +
1
2
)
β
= 180
. Solving for
β
we,
get
β
= 72
o
. Since
4
PBQ is isoceles, the angle
6
QBP
= 32
o
. Now
complete the line BE=AC and the line BD=AC and connect edges AE,
ED and DC. Apply similarity of triangles to show that all edges have
the same length. This completes the proof.
6.3
Regular Polygons
The only regular polygons known to the Greeks were the equilaterial
triangle and the pentagon. It was not until about 1800 that C. F. Guass
added to the list of constructable regular polyons by showing that there
are three more, of 17, 257, and 65,537 sides respectively. Precisely, he
showed that the constructable regular polygons must have
2
m
p
1
p
2
. . . p
r
Pythagoras and the Pythagoreans
25
sides where the
p
1
, . . . , p
r
are distinct
Fermat primes
. A Fermat prime
is a prime having the form
2
2
n
+ 1
.
In about 1630, the Frenchman Pierre de Fermat (1601 - 1665) con-
jectured that all numbers of this kind are prime. But now we know
differently.
Pythagoras and the Pythagoreans
26
Pierre Fermat (1601-1665), was a court
attorney in Toulouse (France). He was an
avid mathematician and even participated in
the fashion of the day which was to recon-
struct the masterpieces of Greek mathemat-
ics. He generally refused to publish, but
communicated his results by letter.
Are there any other Fermat primes? Here is all that is known to date.
It is not known if any other of the Fermat numbers are prime.
p
2
2
p
+ 1
Factors
Discoverer
0
3
3
ancient
1
5
5
ancient
2
17
17
ancient
3
257
257
ancient
4
65537
65537
ancient
5
4,294,957,297 641, 6,700,417
Euler, 1732
6
21
274177,67280421310721
7
39 digits
composite
8
78 digits
composite
9
617 digits
composite
Lenstra, et.al., 1990
10
709 digits
unknown
11
1409 digits
composite
Brent and Morain, 1988
12-20
composite
By the theorem of Gauss, there are constructions of regular poly-
gons of only 3, 5 ,15 , 257, and 65537 sides, plus multiples,
2
m
p
1
p
2
. . . p
r
sides where the
p
1
, . . . , p
r
are distinct
Fermat primes
.
Pythagoras and the Pythagoreans
27
6.4
More Pythagorean Geometry
Contributions
22
by the Pythagoreans include
•
Various theorems about triangles, parallel lines, polygons, circles,
spheres and regular polyhedra. In fact, the sentence in Proclus
about the discovery of the irrationals also attributes to Pythago-
ras the discovery of the five regular solids (called then the ‘cosmic
figures’). These solids, the tetrahedron (4 sides, triangles), cube (6
sides, squares, octahedron (8 sides, triangles), dodecahedron (12
sides, pentagons), and icosahedron (20 sides, hexagons) were pos-
sibly known to Pythagoras, but it is unlikely he or the Pythagoreans
could give rigorous constructions of them. The first four were as-
sociated with the four elements, earth, fire, air, and water, and
because of this they may not have been aware of the icosahedron.
Usually, the name Theaetetus is associated with them as the math-
ematician who proved there are only five, and moreover, who gave
rigorous constructions.
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
•
Work on a class of problems in the applications of areas. (e.g. to
construct a polygon of given area and similar to another polygon.)
•
The geometric solutions of quadratics. For example, given a line
segment, construct on part of it or on the line segment extended a
parallelogram equal to a given rectilinear figure in area and falling
22
These facts generally assume a knowledge of the Pythagorean Theorem, as we know it.
The level of rigor has not yet achieved what it would become by the time of Euclid
Pythagoras and the Pythagoreans
28
short or exceeding by a parallelogram similar to a given one. (In
modern terms, solve
b
c
x
2
+
ax
=
d
.)
6.5
Other Pythagorean Geometry
We know from from Eudemus that the Pythagoreans discovered the
result that the sum of the angles of any triangle is the sum of two right
angles. However, if Thales really did prove that every triangle inscribed
in a right triangle is a right triangle,
he surely would have noted the result
for right triangles. This follows directly
from observing that the base angles of
the isosceles traingles formed from the
center as in the figure just to the right.
The proof for any triangle follows
directly. However, Eudemus notes
A
B
C
O
a different proof. This proof requires the “alternating interior angles”
theorem. That is:
Theorem.
(Euclid,
The Elements
Book
I, Proposition 29.)
A straight line
falling on parallel straight lines make
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.
A
B
C
D
E
From this result and the figure just above, note that the angles
/
ABD
=
/
CAB
and /
CBE
=
/
ACB
. The result follows.
The quadrature of certain
lunes
(crescent shaped regions) was
performed by
Hippocrates of Chios
. He is also credited with the
arrangement of theorems in an order so that one may be proved from a
previous one (as we see in Euclid).
Pythagoras and the Pythagoreans
29
B
A
Lune
C
D
We wish to determine the area of the lune
ABCD
, where the large
segment
ABD
is similar to the smaller segment (with base on one leg
of the right isosceles triangle
4
ABC
). Because segments are to each
other as the squares upon their bases, we have the
Proposition:
The area of the large lune
ABCD
is the area of the triangle
4
ABC
.
This proposition was among the first that determined the area of a curvi-
linear figure in terms of a rectilinear figure. Quadratures were obtained
for other lunes, as well. There resulted great hope and encouragement
that the circle could be squared. This was not to be.
7
The Pythagorean Theory of Proportion
Besides discovering the five regular solids, Pythagoras also discovered
the theory of proportion. Pythagoras had probably learned in Babylon
the three basic means, the
arithmetic
, the
geometric
, and the
subcon-
trary
(later to be called the
harmonic
).
Beginning with
a > b > c
and denoting
b
as the
—mean
of
a
and
c
, they are:
1
a
−
b
b
−
c
=
a
a
arithmetic
a
+
c
= 2
b
Pythagoras and the Pythagoreans
30
2
a
−
b
b
−
c
=
a
b
geometric
ac
=
b
2
3
a
−
b
b
−
c
=
a
c
harmonic
1
a
+
1
c
=
2
b
The most basic fact about these proportions or means is that if
a > c
,
then
a > b > c
. In fact, Pythagoras or more probably the Pythagore-
ans added seven more proportions. Here is the complete list from the
combined efforts of Pappus and Nicomachus.
Formula
Equivalent
4
a
−
b
b
−
c
=
c
a
a
2
+
c
2
a
+
c
=
b
5
a
−
b
b
−
c
=
c
b
a
=
b
+
c
−
c
2
b
6
a
−
b
b
−
c
=
b
a
c
=
a
+
b
−
a
2
b
7
a
−
c
b
−
c
=
a
c
c
2
= 2
ac
−
ab
8
a
−
c
a
−
b
=
a
c
a
2
+
c
2
=
a
(
b
+
c
)
9
a
−
c
b
−
c
=
b
c
b
2
+
c
2
=
c
(
a
+
b
)
10
a
−
c
a
−
b
=
b
c
ac
−
c
2
=
ab
−
b
2
11
a
−
c
a
−
b
=
a
b
a
2
= 2
ab
−
bc
The most basic fact about these proportions or means is that if
a > c
,
then
a > b > c
. (The exception is 10, where b must be selected
depending on the relative magnitudes of
a
and
c
, and in one of the
cases
b
=
c
.) What is very well known is the following relationship
between the first three means. Denote by
b
a
, b
g
,
and
b
h
the arithmetic,
geometric, and harmonic means respectively. Then
b
a
> b
g
> b
h
(1)
The proofs are basic. In all of the statements below equality occurs if
and only if
a
=
c
. First we know that since
(
α
−
γ
)
2
≥
0
, it follows
that
α
2
+
γ
2
≥
2
αγ
. Apply this to
α
=
√
a
and
β
=
√
b
to conclude
Pythagoras and the Pythagoreans
31
that
a
+
c >
2
√
ac
, or
b
a
≥
b
g
. Next, we note that
b
h
= 2
ac
a
+
b
or
b
2
g
=
b
h
b
a
. Thus
b
a
≥
b
g
≥
b
h
.
What is not quite as well known is that the fourth mean, some-
times called the
subcontrary to the harmonic
mean is larger than all
the others except the seventh and the ninth, where there is no greater
than or less than comparison over the full range of
a
and
c
. The proof
that this mean is greater than
b
a
is again straight forward. We easily
see that
b
=
a
2
+
c
2
a
+
c
=
(
a
+
c
)
2
−
2
ac
a
+
c
= 2
b
a
−
b
2
g
b
a
≥
b
a
by (1). The other proofs are omitted.
Notice that the first six of the proportions above are all of a
specific generic type, namely having the form
a
−
b
b
−
c
=
· · ·
. It turns
out that each of the means (the solution for
b
) are comparable. The
case with the remaining five proportions is very much different. Few
comparisons are evident, and none of the proportions are much in use
today. The chart of comparison of all the means below shows a plus
(minus) if the mean corresponding to the left column is greater (less)
than that of the top row. If there is no comparison in the greater or less
than sense, the word “No” is inserted.
Pythagoras and the Pythagoreans
32
i/j
1
2
3
4
5
6
7
8
9
10 11
1
+
+
-
-
-
No No No No +
2
-
+
-
-
-
No No
-
No No
3
-
-
-
-
-
-
No
-
No No
4
+
+
+
+
+ No + No +
+
5
+
+
+
-
+ No + No +
+
6
+
+
+
-
-
No No No No +
7
No No + No No No
No
-
No No
8
No No No
-
-
No No
No +
+
9
No +
+ No No No + No
No No
10
No No No
-
-
No No
-
No
No
11
-
No No
-
-
-
No
-
No No
Comparing Pythagorean Proportions
Linking qualitative or subjective terms with mathematical propor-
tions, the Pythagoreans called the proportion
b
a
:
b
g
=
b
g
:
b
h
the
perfect
proportion. The proportion
a
:
b
a
=
b
h
:
c
was called the
musical
proportion.
8
The Discovery of Incommensurables
Irrationals have variously been attributed to Pythagoras or to the Pythagore-
ans as has their study. Here, again, the record is poor, with much of
it in the account by Proclus in the
4
th
century CE. The discovery is
sometimes given to
Hippasus of Metapontum
(
5
th
cent BCE). One
account gives that the Pythagoreans were at sea at the time and when
Hippasus produced (or made public) an element which denied virtually
all of Pythagorean doctrine, he was thrown overboard. However, later
evidence indicates that Theaetetus
23
of Athens (c. 415 - c. 369 BCE)
23
the teacher of Plato
Pythagoras and the Pythagoreans
33
discovered the irrationality of
√
3
,
√
5
, . . . ,
√
17
, and the dates suggest
that the Pythagoreans could not have been in possession of any sort of
“theory” of irrationals. More likely, the Pythagoreans had noticed their
existence. Note that the discovery itself must have sent a shock to the
foundations of their philosophy as revealed through their dictum
All is
Number
, and some considerable recovery time can easily be surmised.
Theorem
.
√
2
is incommensurable with 1.
Proof.
Suppose that
√
2 =
a
b
, with no common factors. Then
2 =
a
2
b
2
or
a
2
= 2
b
2
.
Thus
24
2
|
a
2
, and hence
2
|
a
. So,
a
= 2
c
and it follows that
2
c
2
=
b
2
,
whence by the same reasoning yields that
2
|
b
. This is a contradiction.
Is this the actual proof known to the Pythagoreans? Note: Unlike
the Babylonians or Egyptians, the Pythagoreans recognized that this
class of numbers was wholly different from the rationals.
“Properly speaking, we may date the very beginnings of “theo-
retical” mathematics to the first proof of irrationality, for in “practical”
(or applied) mathematics there can exist no irrational numbers.”
25
Here
a problem arose that is analogous to the one whose solution initiated
theoretical natural science: it was necessary to ascertain something that
24
The expression
m
|
n
where
m
and
n
are integers means that
m
divides
n
without
remainder.
25
I. M. Iaglom, Matematiceskie struktury i matematiceskoie modelirovanie. [Mathematical
Structures and Mathematical Modeling] (Moscow: Nauka, 1980), p. 24.
Pythagoras and the Pythagoreans
34
it was absolutely impossible to observe (in this case, the incommensu-
rability of a square’s diagonal with its side).
The discovery of incommensurability was attended by the intro-
duction of indirect proof and, apparently in this connection, by the
development of the definitional system of mathematics.
26
In general,
the proof of irrationality promoted a stricter approach to geometry, for it
showed that the evident and the trustworthy do not necessarily coincide.
9
Other Pythagorean Contributions.
The Pythagoreans made many contributions that cannot be described in
detail here. We note a few of them without commentary.
First of all, connecting the concepts of proportionality and relative
prime numbers, the theorem of Archytas of Tarantum (c. 428 - c. 327
BCE) is not entirely obvious. It states that there is no mean proportional
between successive integers. Stated this way, the result is less familiar
than using modern terms.
Theorem.
(Archytas) For any integer
n
, there are no integral solutions
a
to
A
a
=
a
B
where
A
and
B
are in the ratio
n
:
n
+ 1
.
Proof.
The proof in Euclid is a little cumbersome, but in modern
notation it translates into this: Let
C
and
D
be the smallest numbers
in the same ratio as
A
and
B
. That is
C
and
D
are relatively prime.
Let
D
=
C
+
E
Then
C
D
=
C
C
+
E
=
n
n
+ 1
which implies that
Cn
+
C
=
Cn
+
En
. Canceling the terms
Cn
, we
see that
E
divides
C
. Therefore
C
and
D
are not relatively prime, a
contradiction.
26
A. Szabo ”Wie ist die Mathematik zu einer deduktiven Wissenschaft geworden?”, Acta
Antiqua, 4 (1956), p. 130.
Pythagoras and the Pythagoreans
35
The Pythagoreans also demonstrated solutions to special types of
linear systems. For instance, the
bloom
of Thymaridas (c. 350 BCE)
was a rule for solving the following system.
x
+
x
1
+
x
2
+
. . .
+
x
n
=
s
x
+
x
1
=
a
1
x
+
x
2
=
a
2
. . .
x
+
x
n
=
a
n
This solution is easily determined as
x
=
(
a
1
+
. . .
+
a
n
)
−
s
n
−
2
It was used to solve linear systems as well as to solve indeterminate
linear equations.
The Pythagoreans also brought to Greece the earth-centered cos-
mology that became the accepted model until the time of Copernicus
more that two millenia later. Without doubt, this knowledge originated
in Egypt and Babylon. Later on, we will discuss this topic and its
mathematics in more detail.
Pythagoras and the Pythagoreans
36
References
1. Russell, Bertrand,
A History of Western Philosophy
, Simon and
Schuster Touchstone Books, New York, 1945.
