Download Euclid`s Proof of the Pythagorean Theorem

08/19/2015
Math 460 – Senior Seminar
“Let no man ignorant of geometry enter here”
motto across archway of Plato’s Academy
Historical Overview
490 BCE – Battle of Marathon; 480 BCE - Battle of
Thermopylae;
431 – 404 BCE Peloponnesian War
Plato 427 – 347 BCE; Eudoxus 408 – 355 BCE;
Aristotle 384 – 322 BCE
Alexander the Great: 356 – 323 BCE
Alexandra founded – ca. 332 BCE
Euclid’s Elements - 300 BCE
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Eudoxus 408 – 355 BCE
Theory of Proportions
fixed logical scandal of incommensurate quantities
Two quantities a and b are commensurate iff there
is a 3rd quantity c such that a and b are integral
multiples of c.
Method of Exhaustion (proto-integration)
used to compute areas and volumes of nonrectilinear figures
Axiomatic Method
Begin with handful of (self-evident)
axioms/postulates and definitions
Logically develop proofs of propositions based on
axioms, definitions and previously proved
propositions
Establishes logical validity of propositions
Avoids circular arguments
Economy of presupposition
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Ch 2: Euclid’s Proof of the Pythagorean Theorem
Book I Preliminaries
Definitions, Postulates, Common Notions,
The 5th Postulate
The Early Propositions
I.1, I.2, I.5, I.16
Enter Parallelism
I.27, I.29, I.32
Triangles and Parallelograms
I.31, I.35, I.37, I.41
The Proof of I.47
The Five Postulates
1.
2.
3.
4.
5.
[It is possible] to draw a straight line from any point to
any point
[It is possible] to produce a finite straight line
continuously in a straight line
[It is possible] to describe a circle with any center and
distance (i.e. radius)
All right angle are equal to one another
If a straight line meets two straight line so as to make
two interior angles on the same side of it taken together
less than two right angles, the lines if extended shall
meet on that side on which the angles are less than two
right angles
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Early Propositions
I.1
On a given finite straight line, to construct an
equilateral triangle
I.2 From a given point (A) to draw a straight line
equal to another given straight line (BC)
I.3 From the greater of two given straight lines cut
off a part equal to the lesser
I.4 SAS congruency
I.5 The angles at the base of an isosceles triangle are
equal to one another and if the equal sides be
produced, the angles on the other side of the base shall
be equal to one another
I.2 and I.5
I.2
A
I.5
C
B
E
D
A
B
C
F
F
G
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The Constructions
I.9 To bisect an angle
I.10 To bisect a finite line segment
I.11 To construct a perpendicular from any point on
a given line
I.12 . . .or not on a given line
I.31
Given a line and a point not on the line to
construct a second line through the point
parallel to the first
Early Propositions
I.15 Vertical angles are equal
I.16 In any triangle if one of the sides is produced,
then then exterior angle is greater than either of the
interior or opposite angles (used for I.26 AAS
congruence)
F
A
3
1
AE = EC
BE = EF
1 = 2
∴ 3 = 4
E
2
4
B
C
D
I.17 An two angles of a triangle are together less than
two right angles
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An Aside: Saccheri-Legendre Theorem
The sum of the angles of a triangle is less than or equal
to two right angles (i.e. 180 °)
pf (outline): Assume there is a triangle whose angle
sum equals 180° + ε (ε > 0). Given ∆ABC and using the
construction from I.16, construct a 2nd triangle ∆ACE
where either angle ∡EAC or ∡AEC is less than or equal
to ½ ∡ BAC. By repeating this construction one can
eventually obtain a triangle ∆PQR whose angle sum
equals that of ∆ABC but where angle ∡P < ε. Thus
∡Q+∡R > 180° contradicting I.16
Parallelism
I.27 If a straight line falling on two straight lines
makes alternate angles equal to one another, then the
straight lines are parallel to one another
1
2
I.29 A straight line falling on parallel straight lines
makes the alternate angles equal to one another, the
exterior angles equal to the interior and opposite
angle, and the sum of the interior angles on the same
side equal to two right angles.
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Parallelism
I.32 In any triangle if one of the sides is produced
than the exterior angle equals the sum of the two
interior and opposite angles
Triangles and Parallelograms
I.31 To draw a straight line through a given point
parallel to a given straight line.
I.35 Parallelograms which are on the same base and
in the same parallels equal one another
I.37 Triangles which are on the same base and in the
same parallels equal one another.
I.41 If a parallelogram has the same base with a
triangle and is in the same parallels, then the
parallelogram is double the triangle.
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Proof of I.47
I.46 To describe a square on a given straight line.
I.47 In right-angled triangles the square on the side
opposite the right angle equals the sum of the squares
on the sides containing the right angle.
I.48 If in a triangle the square on one of the sides
equals the sum of the squares on the remaining two
sides of the triangle, then the angle contained by the
remaining two sides of the triangle is right.
I.47
H
∆ ≅ ∆
G
K
I.37
A
I.46
F
B
C
I.31
I.41
D
L
E
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Again the 5th Postulate
Proclus’ axiom: If a line intersects one of two parallel
lines it must intersect the other also
Equidistance postulate: Parallel lines are everywhere
equidistant
Playfair’s postulate: Through a point not on a given
line there can be drawn one and only one line parallel
to the given line
The triangle postulate: The sum of the angles of a
triangle is two right angles
The 5th Postulate
Giovanni Girolamo Saccheri (1667-1733)
Saccheri Rectangle
Carl Fredrick Gauss (1777-1855)
Johann Bolyai (1802-1860)
Nikolai Lobachevski ( 1793-1856)
Georg Friedrich Bernhard Riemann (1826-1866)
Eugenio Beltrami (1835-1900)
Non-Euclidian Geometry: AAA is congruence relation
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Saccheri Quadrilateral
AB = CD, base angles ∡B = ∡C are right
Can prove
A
summit angles ∡A = ∡D
EF ⊥ to AD and BC
for midpoints E & F
summit angles ≤ 90°
B
F
D
E
C
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