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Euclids Elements

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.

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.

The ten sections of Chapter 5 concern Euclids

Elements. We will spend two weeks discussing

these sections.

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.

It is known for certain that Euclid was recruited

to teach at the Museum in Alexandria, which is

where he likely completed the Elements.

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.

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.

?

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.

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.

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.

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?

- 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.

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?

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.

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.

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?

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.

What is the difference between the first claim of

Proposition 5.3 and Proposition 5.4?

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?

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.

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

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.

Euclids proof the Pythagorean Theorem,

Proposition 5.14, is believed to be his own. It

makes use of Proposition 5.11.

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.

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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

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

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

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

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

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?

How could you make use of this result to prove

Propositions 5.12 and 5.13?

How could you make use of this result to prove

Propositions 5.12 and 5.13?

How could you make use of this result to prove

Propositions 5.12 and 5.13?

How could you make use of this result to prove

Propositions 5.12 and 5.13?

How could you make use of this result to prove

Propositions 5.12 and 5.13?

How could you make use of this result to prove

Propositions 5.12 and 5.13?

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.

Is the converse of the Pythagorean Theorem true?

Is the converse of the Pythagorean Theorem

true? Yes! Its Proposition 5.16.

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?

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

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

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

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.

(x y)2 x2 2xy y2

x

y

y

x

y

x

x y

y

x

x y

y

x

(x y)2

y

x

x

y

(x y)2

y

x

x

y

(x y)2 x2

y

x

x

x2

y

(x y)2 x2 y2

y

x

x

x2

y

y2

(x y)2 x2 y2 xy

y

x

x

x2

y

y2

xy

(x y)2 x2 y2 xy xy

y

x

x

xy

x2

y

y2

xy

(x y)2 x2 2xy y2

y

x

x

xy

x2

y

y2

xy

(x - y)2 x2 - 2xy y2

x

y

x

x

y

x

y

x

x

y

x - y

x

y

x - y

x

y

x - y

x

y

x - y

x

x

y

x - y

x

x

y

x - y

x - y

x

y

x

(x - y)2

y

x - y

x - y

(x - y)2

x

y

x

(x - y)2 x2 -

y

x - y

x - y

x

x2

y

x

(x - y)2 x2 - xy -

y

x - y

x - y

x

xy

y

y

(x - y)2 x2 - xy - ?

y

x - y

x - y

x

y

?

x

(x - y)2 x2 - xy - ?

y

x - y

x - y

x

y

xy

x

(x - y)2 x2 - xy - ?

y

x - y

x - y

x

y

y2

y

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

y

x - y

x - y

x

y

(xy - y2)

y

(x - y)2 x2 - 2xy y2

y

x - y

x - y

x

y

x

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.

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.

202 lt 740 and 302 gt 740, so cut out a square with

side length 20.

Area is 740.

202 lt 740 and 302 gt 740, so cut out a square with

side length 20.

Area is 740.

20

x

The new square has area 400, and the remaining

gnomon has area 340.

Area is 740.

20

x

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

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

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

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

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

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

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

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

add 0.002 to the side, the remaining area is

Area is 740.

area 1 area 2 area 3 ...

27.202

The square root of 740 is 27.202.

Area is 740.

27.202

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?

Propositions 5.21 and 5.22 describe the

relationships among the squares on the sides of

triangles that are not right.

What does Proposition 5.21 say?

What does Proposition 5.21 say?

B

A

C

What does Proposition 5.21 say?

B

A

C

D

What does Proposition 5.21 say?

B

A

C

D

What does Proposition 5.21 say?

B

A

C

D

What does Proposition 5.21 say?

B

C

D

A

What does Proposition 5.21 say?

B

C

D

A

What does Proposition 5.22 say?

What does Proposition 5.22 say?

A

C

B

What does Proposition 5.22 say?

A

C

B

D

What does Proposition 5.22 say?

A

C

B

D

What does Proposition 5.22 say?

A

C

B

D

What does Proposition 5.22 say?

A

C

B

D

What does Proposition 5.22 say?

A

C

B

D

Lets look at the proof of Proposition 5.26.

Lets look at the proof of Proposition 5.26.

Lets look at the proof of Proposition 5.26.

We will show that

area

equals

area

Lets look at the proof of Proposition 5.26.

Lets look at the proof of Proposition 5.26.

Lets look at the proof of Proposition 5.26.

Lets look at the proof of Proposition 5.26.

Lets look at the proof of Proposition 5.26.

The perpendiculars

Lets look at the proof of Proposition 5.26.

The perpendiculars

Lets look at the proof of Proposition 5.26.

The perpendiculars bisect the chords

Lets look at the proof of Proposition 5.26.

The perpendiculars bisect the chords

Lets look at the proof of Proposition 5.26.

The perpendiculars bisect the chords by

Proposition 5.24.

Lets look at the proof of Proposition 5.26.

This means line AC

Lets look at the proof of Proposition 5.26.

This means line AC

A

C

Lets look at the proof of Proposition 5.26.

This means line AC is cut into equal segments at G

A

C

Lets look at the proof of Proposition 5.26.

This means line AC is cut into equal segments at G

A

G

C

Lets look at the proof of Proposition 5.26.

This means line AC is cut into equal segments at G

A

G

C

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

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

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

Lets look at the proof of Proposition 5.26.

By Proposition 5.19 the rectangle

A

G

B

C

Lets look at the proof of Proposition 5.26.

By Proposition 5.19 the rectangle

A

G

B

C

Lets look at the proof of Proposition 5.26.

By Proposition 5.19 the rectangle and square

A

G

B

C

Lets look at the proof of Proposition 5.26.

By Proposition 5.19 the rectangle and square

A

G

equal the square

B

C

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

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

Lets look at the proof of Proposition 5.26.

By Proposition 5.19 the rectangle and squares

A

equal the squares

G

B

C

Lets look at the proof of Proposition 5.26.

By Proposition 5.19 the rectangle and squares

A

equal the squares

G

B

C

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)

Lets look at the proof of Proposition 5.26.

Similarly the rectangle

Lets look at the proof of Proposition 5.26.

Similarly the rectangle and squares

Lets look at the proof of Proposition 5.26.

Similarly the rectangle and squares

equal the squares

Lets look at the proof of Proposition 5.26.

Similarly the rectangle and squares

equal the squares

which equal the square (Pythagorean Theorem)

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)

Lets look at the proof of Proposition 5.26.

This means

equal

Lets look at the proof of Proposition 5.26.

This means

equal

Both pairs of squares equal by

the Pythagorean Theorem.

Lets look at the proof of Proposition 5.26.

This means

equal

Both pairs of squares equal by

the Pythagorean Theorem.

Lets look at the proof of Proposition 5.26.

This means

equal

Both pairs of squares equal by

the Pythagorean Theorem.

Lets look at the proof of Proposition 5.26.

This means

equal

Lets look at the proof of Proposition 5.26.

So

equals

Lets look at the proof of Proposition 5.26.

So

equals

as claimed.