Transcript Document 7314372

Why Studying Euclidean Geometry?
• It is well known that most students find it difficult to learn
to do proofs. Geometry, especially Euclid, provides an
excellent setting for students to improve their proof skills.
• Secondary school teachers are going to be teaching
Euclidean geometry so they need to know it.
• We believe that a student who does not have a good
background in Euclidean geometry is not in a position to
get the point of non-Euclidean geometry.
Significances of the discovery of
non-Euclidean Geometry
• Most of people are unaware that around a century and a half ago
a revolution took place in the field of geometry that was
scientifically profound as the Copernican revolution in
astronomy and, in its impact, as philosophically important as the
Darwinian theory of evolution.
• The effect of the discovery of hyperbolic geometry on our ideas
of truth and reality has been so profound, writes the great
Canadian geometer H.S.M. Coxeter, “that we can hardly
imagine how shocking the possibility of a geometry different
from Euclid’s must have seemed in 1820.”
• Albert Einstein stated that without this new conception of
geometry, he would not have been able to develop the theory of
relativity.
• Euclidean geometry is the kind of geometry you learned in
high school, the geometry most of us use to visualize the
physical universe.
• It comes from the text by Greek mathematician Euclid, the
Elements, written around 300 B.C.
• Our picture of the physical universe based on this
geometry was painted largely by Isaac Newton in the late
seventeenth century.
• Geometries that differ from Euclid’s own arose out of a
deeper study of parallelism.
• Consider this diagram of two rays perpendicular to
segment PQ:
P
Q
• In Euclidean geometry the perpendicular distance between
the rays remains equal to the distance from P to Q as we
move to the right.
• However, in the early nineteenth century two alternative
geometries were proposed. In hyperbolic geometry the
distance between the rays increases.
• In elliptic geometry the distance decreases and the rays
eventually meet.
• These non-Euclidean geometries were later incorporated in
a much more general geometry developed by C.F. Gauss
and G. F. B. Riemann.
• Now, we continue with a brief history of geometry in
ancient times, and emphasize the development of the
axiomatic method by the Greeks. And finally we will
present Euclid’s five postulates.
A Quick Review of the History of
Euclidean Geometry
Plato 427-347 BCE
Euclidean 325-265 BCE
Let no one ignorant of geometry enter this door.
Entrance to Plato’s Academy
• The word “geometry” comes from the Greek geometrien
(geo: earth, and metrein: to measure); geometry was
originally the science of measuring land.
• The Greek historian Herolotus (5th century B.C.) credits
Egyptian surveyors with having originated the subject of
geometry, but other ancient civilizations (Babylonian,
Hindu, Chinese) also possessed much geometric
information.
• Egyptian geometry was not a science in the Greek sense,
only a grab bag of rules for calculation without any
motivation or justification.
• The Babylonians were much more advanced than the
Egyptians in arithmetic and algebra. Moreover, they knew
the Pythagorean theorem – in a right triangle the square of
the length of the hypotenuse is equal to the sum of the
squares of the lengths of the legs – long before Pythagoras
was born.
• However, the Greeks, beginning with Thales of Milete,
insisted that geometric statements be established by
deductive reasoning rather than that by trial and error.
• The orderly development of theorems by proof was
characteristic of Greek mathematics and entirely new.
• The systematization begun by Thales was continued over
the next two centuries by Pythagoras and his disciples.
• Pythagoras was regarded as a religious prophet. The
Pythagoreans differed from other religious sects in their
belief that elevation of the soul and union with God are
achieved by the study of music and mathematics.
• The Pythagoreans were greatly shocked when they
discovered irrational lengths, such as 2 .
Pythagoras of Samos
569BC - 475BC
Thales of Miletus
624BC - 547BC
• Later, Plato repeatedly
cited
the proof for the irrationality
1862
- 1943
of the length of a diagonal of the unit square as an
illustration of the method of indirect proof.
• The point is that this irrationality of length could never
have been discovered by physical measurements, which
always include a small experimental margin of error.
• Hilbert's work in geometry had the greatest influence in that area
after Euclid. A systematic study of the axioms of Euclidean
geometry led Hilbert to propose 21 such axioms and he analysed
their significance. He published Grundlagen der Geometrie in
1899 putting geometry in a formal axiomatic setting. The book
continued to appear in new editions and was a major influence in
promoting the axiomatic approach to mathematics which has
been one of the major characteristics of the subject throughout
the 20th century.
• Hilbert's famous 23 Paris problems challenged (and still
today challenge) mathematicians to solve fundamental
questions. Hilbert's famous speech The Problems of
Mathematics was delivered to the Second International
Congress of Mathematicians in Paris. It was a speech
full of optimism for mathematics in the coming century
and he felt that open problems were the sign of vitality
in the subject.
We must know, we shall know
•
A. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrödinger, E. Verschaffelt,
W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin,
P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M.
Born, N. Bohr,
I. Langmuir, M. Planck, Mme. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R.
Wilson, O.W. Richardson
Chapter 0 : Notations and Conventions
• We will introduce our basic terminology for geometric
objects and the relations between them.
• Our main concern is the theorems of Euclidean Geometry
rather than the abstract construction of this geometry as an
axiom system.
• Points : will be denoted by capital letters A,B,…
• Lines : the line containing A and B will be denoted by AB ;
however sometimes we will use script letters like  ,…
• Three or more points that lie on the same line are said to be
collinear.
• Three or more lines which meet at the same point are said
to be concurrent at that point.
• A line divides the plane into two half planes, each of
which may be described by specifying the line and any one
point in the half plane.
• Rays (half lines) : The ray with initial point A and
containing point B will be denoted AB .
• Line Segments : The line segment with end points A and
B will be denoted AB ; and the length of AB will be
denoted AB.
• A point P on AB that is distinct from A and B is called an
interior point of AB . P is the midpoint of AB if PA  PB.
• Angles : The angle with vertex A and sides AB and AC
will be denoted  A or BAC . The same symbols are also
used to denote the sizes of angles.
• We will use degrees rather than radians for angle
measurements.
• A right angle is one whose measurement is 90; an acute
angle has measure less than 90 ; while an obtuse angle has
measure greater than 90 .
• If A  B  90 then these angles are said to be
complements of each other, whereas they are called
supplements of each other if their sum is 180.
• The interior of ABC is the set of points common to the
half plane on the C side of AB and the A side of BC ,
excluding the points on BA and BC .
• When P is interior to ABC , then BP is a bisector of this
angle if ABP  PBC .
• Congruence : Two line segments of the same length are
said to be congruent.AB is congruent to CD is shown by
AB  CD
• Note that AB  CD is not the same as AB  CD, which means
that AB and CD are the same line segment.
• We also say that two angles are congruent if they have the
same size. In this case A  B and A  B
mean the same thing.
• Triangles : The triangle with vertices A, B, and C is
denoted ABC .
• An equilateral triangle has all three sides the same length.
An obtuse triangle has an obtuse angle at some vertex,
where as an acute triangle has no obtuse or right angles.

• A right triangle has a 90 angle. The side opposite this
angle is called the hypotenuse.
The Axiomatic Method
• Mathematicians can make use of trial and error, computation
of special cases, inspired guessing, or any other way to
discover theorems. The axiomatic method is a method of
proving that results are correct.
• So proofs give us assurance that results are correct. In many
cases they also give us more general results.
• For example, the Egyptians and Hindus knew by experiment
that if a triangle has sides of lengths 3, 4, and 5, it is a right
triangle.
• But the Greeks proved that if a triangle has sides of lengths a,
b, and c, and if a 2  b 2  c 2, then the triangle is a right triangle.
• It would take an infinite number of experiments to check this
result (and, besides, experiments only measure things
approximately).
• There are two requirements that must be met for us to agree
that a proof is correct:
• REQUIREMENT 1. Acceptance of certain statements called
“axioms,”, or “postulates” without further justification.
• REQUIREMENT 2. Agreement on how and when one
statement “follows logically” from another, i.e., agreement on
certain rules of reasoning.
• Euclid’s monumental achievement was to single out a few
simple postulates, statements that were acceptable without
further justification, and then to deduce from them 465
propositions, many complicated and not at all intuitively
obvious, which contained all the geometric knowledge of his
time.
• One reason the Elements is such a beautiful work is that so
much has been deduced from so little.
• We have been discussing what is required for us to agree that a
proof is correct. Here is one requirement that we took for
granted:
• REQUIREMENT 0: Mutual understanding of the meaning of
the words and symbols used in the discourse.
• If, for example, I define a right angle to be a 90 angle, and

then define a 90 angle to be a right angle, I would violate the
rule against circular reasoning.
•
Also, we cannot define every term that we use. In order to
define one term we must use other terms, and to define
these terms we must use still other terms, and so on. If we
were not allowed to leave some terms undefined, we
would get involved in infinite regress.
•
To apply the Axiomatic System Method in geometry, we
have to consider the following objects:
Undefined Terms or Primaries
Defined Terms
Axioms or Postulates
Logic or Rules of Reasoning
Theorems
1.
2.
3.
4.
5.
• Here are the five undefined geometric terms that are the
basis for defining all other geometric terms in plane
Euclidean geometry:
• Point
• Line
• Lie on (as “two points lie on a unique line”)
• Between (as in “point C is between points A and B”)
• Congruent
A
C
M
B
D
l
• The mentioned list of undefined geometric terms is due to
David Hilbert (1862 - 1943). His treatise, The
Foundations of Geometry (1899), not only clarified
Euclid’s definitions but also filled in the gaps in some of
Euclid’s proofs.
• Some other mathematicians who worked to establish
rigorous foundations for Euclidean geometry are: G.
Peano, M. Pieri, G. Veronese, O. Veblen, G. de B.
Robinson, E. V. Huntington, and H. G. Forder.
• These mathematicians used lists of undefined terms
different from the one used by Hilbert. Pieri used only two
undefined terms (as a result, however, his axioms were
more complicated).
Euclid’s First Four Postulates
• Euclid based his geometry on five fundamental
assumptions, called axioms or postulates.
• EUCLID’S POSTULATE I. For every point P and for
every point Q not equal to P there exists a unique line
that passes through P and Q.
• EUCLID’S POSTULATE II. For every segment AB and
for every segment CD there exists a unique point E such
that B is between A and E and segment CD is congruent to
segment BE.
A
C
D
B
E
• EUCLID’S POSTULATE III. For every point O and every
point A not equal to O there exists a circle with center O and
radius OA.
O
A
• EUCLID’S POSTULATE IV. All right angles are congruent to
each other.
• Euclid’s first postulates have always been readily accepted by
mathematicians. The fifth (parallel) postulate, however, was
highly controversial until the nineteenth century.
• In fact, consideration of alternatives to Euclid’s parallel postulate
resulted in the development of non-Euclidean geometries.
• THE EUCLIDEAN PARALLEL POSTULATE. For
every line  and for every point P that does not lie on 
there exists a unique line m through P that is parallel to .
P
m
l
• Why should this postulate be so controversial?
• It may seem “obvious” to you, perhaps because you have
been conditioned to think in Euclidean terms.
• However, if we consider the axioms of geometry as
abstractions from experience, we can see a difference
between this postulate and the other four.
• The first two postulates are abstractions from our
experiences drawing with a straightedge; the third
postulate derives from our experiences drawing with a
compass. The fourth postulate also derives from our
experiences measuring angles with a protractor.
• The fifth postulate is different in that we cannot verify
empirically whether two lines meet, since we can draw
only segments, not lines.
Attempts to Prove the Parallel Postulate
• Euclid himself recognized the questionable nature of the
parallel postulate, for he postponed using it for as long as
he could.
• Remember that an axiom was originally supposed to be so
simple and intuitively obvious that no one could doubt its
validity.
• From the very beginning, however, the parallel postulate
was attacked as insufficiently plausible to qualify as an
unproved assumption.
• For two thousand years mathematicians tried to derive it
from the other four postulates or to replace it with another
postulate, one more self-evident.
• All attempts to derive it from the first four postulates
turned out to be unsuccessful because the so-called proofs
always entailed a hidden assumption that was unjustifiable.
• The substitute postulates, purportedly more self-evident,
turned out to be logically equivalent to the parallel
postulate, so that nothing was gained by the substitution.