Euclids Elements

1
Euclids Elements
2
Plato on mathematicians
And do you not know also that although they
further make use of the visible forms and reason
about them, they are thinking not of these, but
of the ideals which they resemblebut they are
really seeking to behold the things themselves,
which can be seen only with the eye of the mind.
3
Proclus on mathematics
We have learned from the very pioneers of this
science not to have any regard to mere plausible
imaginings when it is a question of the
reasonings to be included in our geometrical
doctrine.
4
The ten sections of Chapter 5 concern Euclids
Elements. We will spend two weeks discussing
these sections.
5
Few facts about Euclid are known with certainty.
It is likely that he studied at Platos Academy.
There he would have learned Aristotles method of
proof and reasoning, which features prominently
in his Elements, as well as the original
geometric ideas of Theaetetus and Eudoxus, which
also appear. He may also have studied earlier
compilations of known geometry, such as
Hippocratess Elements, that served as
inspiration for his own Elements.
6
It is known for certain that Euclid was recruited
to teach at the Museum in Alexandria, which is
where he likely completed the Elements.
7
What is Euclids Elements? An element is a
fundamental theorem, as Aristotle explains We
give the name elements to those geometrical
propositions the proof of which are implied in
the proof of all or most of the others. Proclus
explains the Elements with an analogy. He says
Euclids elements are to geometry as the letters
of the alphabet are to language.
8
The purpose of Euclids Elements is subject to
debate. Since it omits advanced results on
conics and spherical geometry, some believe it
was less of a text for established mathematicians
and more of an introductory text for
students. The impact of the Elements, on the
other hand, is unquestionable. It is succeeded
only by the Bible in the number of translations,
editions, and commentaries since its first
printing, and has profoundly influenced some of
historys greatest minds.
?
9
The purpose of Euclids Elements is subject to
debate. Since it omits advanced results on
conics and spherical geometry, some believe it
was less of a text for established mathematicians
and more of an introductory text for
students. The impact of the Elements, on the
other hand, is unquestionable. It is succeeded
only by the Bible in the number of translations,
editions, and commentaries since its first
printing, and has profoundly influenced some of
historys greatest minds.
10
The influence of Euclids Elements can be found
in Isaac Newtons Principia and Kants Critique
of Pure Reason. Abraham Lincoln mastered the
first six books to improve his reasoning skills,
and Albert Einstein described Euclidean geometry
as a second wonder in his life. Two thousand
years later, students still study Euclidean
geometry in school.
11
The Elements culminated the classical Greek
tradition of theoretical mathematics. Euclid
recognized, compiled, and carefully arranged all
important geometry known at the time into
thirteen volumes. Each result included in the
Elements is deduced from those which precede it.
Euclid had to analyze all known geometry in order
to determine which results depended on which
others and put them all in the proper order.
12
Since one cannot carry a chain of logical
reasoning backwards indefinitely (result A
depends on result B which relies on C, and so
on), Euclid had to start somewhere. He organized
his chain of logic so that the starting point was
as simple and intuitive as possible (see page
90). Do you find anything unexpected in Euclids
list of postulates?
13

• Statements logically equivalent to Eulids fifth
  postulate
• Exactly one line can be drawn through any point
  not on a given line parallel to the given line.
• The sum of the angles in every triangle is 180.
• There exists a triangle whose angles add up to
  180.
• The sum of the angles is the same for every
  triangle.
• There exists a pair of similar, but not
  congruent, triangles.
• Every triangle can be circumscribed.
• If three angles of a quadrilateral are right
  angles, then the fourth angle is also a right
  angle.
• There exists a quadrilateral of which all angles
  are right angles.
• There exists a pair of straight lines that are at
  constant distance from each other.
• Two lines that are parallel to the same line are
  also parallel to each other.
• Given two parallel lines, any line that
  intersects one of them also intersects the other.
  
• In a right-angled triangle, the square of the
  hypotenuse equals the sum of the squares of the
  other two sides.
• There is no upper limit to the area of a
  triangle.

14
Although he doesnt explicitly say so in his
postulates, Euclid later assumes that the
straight line that many be drawn between any two
points is unique, as is the indefinite extension
of a given straight line. On page 91 we find the
list of the common notions Euclid included with
the postulates at the beginning of the Elements.
What is the difference between a postulate and a
common notion?
15
Just as one cannot continue a chain of logical
reasoning indefinitely backward, so too must one
have a starting point for the definition of
terms. In modern times we present Euclidean
geometry beginning with the terms point, line,
and so on, left undefined. Euclid may have
attempted to define these terms as described on
pages 87 88, but these definitions are so
different in character from those which follow
that some suspect Euclid may indeed have left
them undefined. Since no original manuscripts
exist, we cannot know for sure what the original
Elements contained.
16
The ten sections in Chapter 5 provide a survey of
the thirteen books that compose the Elements. We
will work through several of the results to gain
both a sense of what appears in the work and an
appreciation for the elegance of the proofs.
17
Proposition 5.1 on page 93 shows us how to
construct an equilateral triangle starting with
one side. What familiar theorem is expressed by
Proposition 5.2?
18
Proposition 5.1 on page 93 shows us how to
construct an equilateral triangle starting with
one side. What familiar theorem is expressed by
Proposition 5.2? One of the homework problems
asks you to prove the SSS result for triangles.
Model your proof after that for Proposition 5.2.
You will also need the previous homework problem.
When you do the homework for this section, try
to rely only on results that have been
established so far.
19
What is the difference between the first claim of
Proposition 5.3 and Proposition 5.4?
20
What is the difference between the first claim of
Proposition 5.3 and Proposition 5.4? Euclid
began Book I by proving as many theorems as
possible without relying on the fifth postulate.
Why might he have done this?
21
Euclids fifth postulate was so much longer and
more complicated that the others, that many
mathematicians became convinced they could prove
it from the first four. In other words, it
wasnt really an independent idea, but rather
could be derived from the first four
postulates. It was in trying to derive a
contradiction by assuming the fifth postulate
false that the non-Euclidean geometries were
discovered.
22
Section 5.3 introduces Euclids treatment of
parallel lines. In order to avoid use of the
fifth postulate for as long as possible, he
introduced Proposition 5.7. What does it say for
the triangle below?
1
2
3
4
12
5
8
9
6
10
7
11
23
With this result Euclid proved the familiar
Proposition 5.8 without relying on the fifth
postulate. The converse to this proposition
together with an additional result appears as
Proposition 5.9.
24
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11.
25
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. This link
demonstrates the result.
26
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
27
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
28
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
29
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
30
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
31
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
32
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
33
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
34
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
35
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
36
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
37
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
38
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
39
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
40
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
41
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
42
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
43
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
44
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
45
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
46
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
47
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
48
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
49
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
50
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
2
3
1
4
51
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
2
3
1
4
area 1 area 2 area 2 area 3
52
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
2
3
1
4
area 1 area 3
53
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
2
3
1
4
area 1 area 4 area 3 area 4
54
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
2
3
1
4
area 1 area 4 area 3 area 4
55
Euclids proof the Pythagorean Theorem,
Proposition 5.14, is believed to be his own. It
makes use of Proposition 5.11. How did Euclid
prove it?
56
How could you make use of this result to prove
Propositions 5.12 and 5.13?
57
How could you make use of this result to prove
Propositions 5.12 and 5.13?
58
How could you make use of this result to prove
Propositions 5.12 and 5.13?
59
How could you make use of this result to prove
Propositions 5.12 and 5.13?
60
How could you make use of this result to prove
Propositions 5.12 and 5.13?
61
How could you make use of this result to prove
Propositions 5.12 and 5.13?
62
Euclids proof of the Pythagorean Theorem also
requires Proposition 5.14. Complements about the
diameter are defined on page 98. This link
demonstrates the main idea of Euclids proof.
More precisely Euclid moved triangles (rather
than rectangles) along parallel lines. This link
demonstrates Euclids actual proof. Here are
some other proofs I like 1, 2, 3, 4.
63
Is the converse of the Pythagorean Theorem true?
64
Is the converse of the Pythagorean Theorem
true? Yes! Its Proposition 5.16.
65
Section 5.4 gives an overview of Book II of the
Elements. It gives familiar algebraic identities
expressed geometrically. What algebraic
identities do Propositions 5.17, 5.18, and 5.19
express?
66
Section 5.4 gives an overview of Book II of the
Elements. It gives familiar algebraic identities
expressed geometrically. What algebraic
identities do Propositions 5.17, 5.18, and 5.19
express? x(y z w) xy xz xw
67
Section 5.4 gives an overview of Book II of the
Elements. It gives familiar algebraic identities
expressed geometrically. What algebraic
identities do Propositions 5.17, 5.18, and 5.19
express? x(y z w) xy xz xw (x y)2
x2 y2
68
Section 5.4 gives an overview of Book II of the
Elements. It gives familiar algebraic identities
expressed geometrically. What algebraic
identities do Propositions 5.17, 5.18, and 5.19
express? x(y z w) xy xz xw (x y)2
x2 y2 (x y)(x y)x2 y2
69
The following slides demonstrate how the ancient
Greeks understood algebra from a geometric point
of view. Several of the homework exercises ask
you to prove algebraic identities geometrically.
70
(x y)2 x2 2xy y2
71
x
y
72
y
x
73
y
x
74
x y
y
x

75
x y
y
x

76
(x y)2
y
x

x

y
77
(x y)2
y
x
x
y
78
(x y)2 x2
y
x

x
x2

y
79
(x y)2 x2 y2
y
x

x
x2

y
y2
80
(x y)2 x2 y2 xy
y
x

x
x2

y
y2
xy
81
(x y)2 x2 y2 xy xy
y
x

x
xy
x2

y
y2
xy
82
(x y)2 x2 2xy y2
y
x

x
xy
x2

y
y2
xy
83
(x - y)2 x2 - 2xy y2
84
x
y
85
x
x
y
86
x
y
x
87
x
y
x - y
88
x
y
x - y
89
x
y
x - y
90
x
y
x - y
x
91
x
y
x - y
x
x
92
y
x - y
x - y
x
y
x
93
(x - y)2
y
x - y
x - y
(x - y)2
x
y
x
94
(x - y)2 x2 -
y
x - y
x - y
x
x2
y
x
95
(x - y)2 x2 - xy -
y
x - y
x - y
x
xy
y
y
96
(x - y)2 x2 - xy - ?
y
x - y
x - y
x
y
?
x
97
(x - y)2 x2 - xy - ?
y
x - y
x - y
x
y
xy
x
98
(x - y)2 x2 - xy - ?
y
x - y
x - y
x
y
y2
y
99
(x - y)2 x2 - xy - (xy - y2)
y
x - y
x - y
x
y
(xy - y2)
y
100
(x - y)2 x2 - 2xy y2
y
x - y
x - y
x
y
x
101
Section 5.4 also describes a method for
approximating square roots. The Greeks would
have done this in base 60. We will follow the
procedure described by Example 5.1 in base 10.
102
To approximate the square root of 740, we suppose
have a square with area 740, and approximate the
length of its side.
Area is 740.
103
202 lt 740 and 302 gt 740, so cut out a square with
side length 20.
Area is 740.
104
202 lt 740 and 302 gt 740, so cut out a square with
side length 20.
Area is 740.
20
x
105
The new square has area 400, and the remaining
gnomon has area 340.
Area is 740.
20
x
106
The new square has area 400, and the remaining
gnomon has area 340.
Area is 740.
area 1 area 2 area 3 340
20
2
3
x
1
107
area 1 area 2 20x, so 220x lt 340 try x 8
area 1 area 2 area 3 220(8) 82 384 x
8 is too large
Area is 740.
area 1 area 2 area 3 340
20
2
3
x
1
108
try x 7 220(7) 72 329 add 7 to the side,
the remaining area is 340 329 11
Area is 740.
area 1 area 2 area 3 340
20
2
3
x
1
109
try x 7 220(7) 72 329 add 7 to the side,
the remaining area is 340 329 11
Area is 740.
area 1 area 2 area 3 11
20
2
7
x
1
3
110
area 1 area 2 27x, so 227x lt 11 try x 0.2
area 1 area 2 area 3 227(0.2) (0.2)2
10.84
Area is 740.
area 1 area 2 area 3 11
20
2
7
x
1
3
111
add 0.2 to the side, the remaining area is 11
10.84 0.16.
Area is 740.
area 1 area 2 area 3 11
20
2
7
x
1
3
112
area 1 area 2 (27.2)x, so 2(27.2)x lt
0.16 try x 0.003 area 1 area 2 area 3
2(27.2)(0.002) (0.002)2 gt 0.16
Area is 740.
area 1 area 2 area 3 0.16
27.2
113
area 1 area 2 (27.2)x, so 2(27.2)x lt
0.16 try x 0.002 area 1 area 2 area 3
2(27.2)(0.002) (0.002)2 lt 0.16
Area is 740.
area 1 area 2 area 3 0.16
27.2
114
add 0.002 to the side, the remaining area is
Area is 740.
area 1 area 2 area 3 ...
27.202
115
The square root of 740 is 27.202.
Area is 740.
27.202
116
A previous homework problem asked you to find the
numerical value for the section. The ancient
Greeks would not have viewed the section as a
number, but rather, as a length. Proposition
5.20 shows how to construct this length with
straightedge and compass. What algebraic
equation does this correspond to solving?
117
Propositions 5.21 and 5.22 describe the
relationships among the squares on the sides of
triangles that are not right.
118
What does Proposition 5.21 say?
119
What does Proposition 5.21 say?
B
A
C
120
What does Proposition 5.21 say?
B
A
C
D
121
What does Proposition 5.21 say?
B
A
C
D
122
What does Proposition 5.21 say?
B
A
C
D
123
What does Proposition 5.21 say?
B
C
D
A
124
What does Proposition 5.21 say?
B
C
D
A
125
What does Proposition 5.22 say?
126
What does Proposition 5.22 say?
A
C
B
127
What does Proposition 5.22 say?
A
C
B
D
128
What does Proposition 5.22 say?
A
C
B
D
129
What does Proposition 5.22 say?
A
C
B
D
130
What does Proposition 5.22 say?
A
C
B
D
131
What does Proposition 5.22 say?
A
C
B
D
132
Lets look at the proof of Proposition 5.26.
133
Lets look at the proof of Proposition 5.26.
134
Lets look at the proof of Proposition 5.26.
We will show that
area
equals
area
135
Lets look at the proof of Proposition 5.26.
136
Lets look at the proof of Proposition 5.26.
137
Lets look at the proof of Proposition 5.26.
138
Lets look at the proof of Proposition 5.26.
139
Lets look at the proof of Proposition 5.26.
The perpendiculars
140
Lets look at the proof of Proposition 5.26.
The perpendiculars
141
Lets look at the proof of Proposition 5.26.
The perpendiculars bisect the chords
142
Lets look at the proof of Proposition 5.26.
The perpendiculars bisect the chords
143
Lets look at the proof of Proposition 5.26.
The perpendiculars bisect the chords by
Proposition 5.24.
144
Lets look at the proof of Proposition 5.26.
This means line AC
145
Lets look at the proof of Proposition 5.26.
This means line AC
A
C
146
Lets look at the proof of Proposition 5.26.
This means line AC is cut into equal segments at G
A
C
147
Lets look at the proof of Proposition 5.26.
This means line AC is cut into equal segments at G
A
G
C
148
Lets look at the proof of Proposition 5.26.
This means line AC is cut into equal segments at G
A
G
C
149
Lets look at the proof of Proposition 5.26.
This means line AC is cut into equal segments at
G and unequal segments at B.
A
G
C
150
Lets look at the proof of Proposition 5.26.
This means line AC is cut into equal segments at
G and unequal segments at B.
A
G
B
C
151
Lets look at the proof of Proposition 5.26.
This means line AC is cut into equal segments at
G and unequal segments at B.
A
G
B
C
152
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle
A
G
B
C
153
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle
A
G
B
C
154
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle and square
A
G
B
C
155
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle and square
A
G
equal the square
B
C
156
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle and square
A
equal the square
G
B
C
Add the square to both
157
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle and square
A
equal the square
G
B
C
Add the square to both
158
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle and squares
A
equal the squares
G
B
C
159
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle and squares
A
equal the squares
G
B
C
160
Lets look at the proof of Proposition 5.26.
By Proposition 5.19 the rectangle and squares
A
equal the squares
G
B
C
which equal the square (Pythagorean Theorem)
161
Lets look at the proof of Proposition 5.26.
Similarly the rectangle
162
Lets look at the proof of Proposition 5.26.
Similarly the rectangle and squares
163
Lets look at the proof of Proposition 5.26.
Similarly the rectangle and squares
equal the squares
164
Lets look at the proof of Proposition 5.26.
Similarly the rectangle and squares
equal the squares
which equal the square (Pythagorean Theorem)
165
Lets look at the proof of Proposition 5.26.
Similarly the rectangle and squares
equal the squares
Equal to the previous square since radii are
equal.
which equal the square (Pythagorean Theorem)
166
Lets look at the proof of Proposition 5.26.
This means
equal
167
Lets look at the proof of Proposition 5.26.
This means
equal
Both pairs of squares equal by
the Pythagorean Theorem.
168
Lets look at the proof of Proposition 5.26.
This means
equal
Both pairs of squares equal by
the Pythagorean Theorem.
169
Lets look at the proof of Proposition 5.26.
This means
equal
Both pairs of squares equal by
the Pythagorean Theorem.
170
Lets look at the proof of Proposition 5.26.
This means
equal
171
Lets look at the proof of Proposition 5.26.
So
equals
172
Lets look at the proof of Proposition 5.26.
So
equals
as claimed.