Download MS2013: Euclidean Geometry

MS2013: Euclidean Geometry
Anca Mustata
January 10, 2012
Warning: please read this text with a pencil at hand,
as you will need to draw your own pictures to illustrate
some statements.
Euclid’s Geometry as a Mathematical Theory
”God is always doing geometry”, words attributed to
Plato by Plutarch, suggest the reverence with which
this branch of mathematics was regarded by thinkers in
the ancient world. They saw geometry as managing to
extract proportions, order and symmetry from the seemingly chaotic nature, thus making its beauty accessible
to the reasoning mind. For us as well, geometry is a
bridge from visual representations of the world to abstract logical thinking. This makes it a wonderful education tool. Indeed, since our perception of the world is
embedded in sensorial experiences, what better way to
develop a solid basis for our abstract thinking than to
combine our visual intuition with logical deductions?
It is for these reasons that a an ancient geometry text
has been referred to as the most famous and influential
textbook ever written. The Elements is a collection of
thirteen mathematical books attributed to Euclid, who
taught at Alexandria in Egypt and lived from about 325
BC to 265 BC. This is the earliest known historical example of a mathematical theory based on the axiomatic
and logical deduction method.
A mathematical theory consists of
• a set of basic objects described by Definitions,
• a set of basic assumptions about these objects,
called Axioms, and
• a set of statements derived from the axioms by
logical reasoning.
– The most important of these statements are
called Theorems,
– less important statements are called Propositions,
– Corrolaries are direct consequences of some
previous statement, and
– Lemmas are helpful in proving further propositions or theorems.
Each theorem, proposition or lemma consists of
• a Hypothesis (set of assumptions), which is what
We Know, and
• a Conclusion, which is what we have To Prove.
• These should be followed by a Proof, meaning a
chain of statements related by logical implications,
which starts from the hypothesis, combines it with
the axioms and/or statements already proven, and
arrives to the conclusion.
Basic objects and terminology of Euclidean geometry
”All human knowledge begins with intuitions, thence
passes to concepts and ends with ideas.” (Kant, Kritik
der reinen Vernunft, Elementarlehle).
Euclid’s geometry assumes an intuitive grasp of basic
objects like points, straight lines, segments, and the
plane. These could be considered as primitive concepts,
in the sense that they cannot be described in terms of
simpler concepts. However, Euclid’s Elements do attempt some definitions by means of other intuitive notions like position, breath and length, in-between. Some
of these definitions are included below in italics for your
enjoyment; you do not need to remember them, but
you do need to know the definitions of more complex
objects like the circle or a polygon, or relations between
them like concurrence, perpendicularity, etc.
The Point A point is that of which there is no part.
A point is usually denoted by an upper case letter.
The Line A straight line is length without breadth,
which lies evenly with points on itself.
A straight line is usually denoted by a lower case letter.
We will think of a line as a set of points. We write
A ∈ d if A is a point on the line d. Alternatively, we
may denote a line by any two points on it: d = AB .
d
A
b
B
b
Note: We will call any straight line shortly ”line”. Even
though we are forced to draw a line in a finite space,
we should think of it as extending forever both on
the left and the right hand side.
A point A on a line d divides the line into two half-lines,
or rays.
Two points A and B on the line d determine the segment [AB] = AB, made of all the points between A and
B.
b
A
[AB]
b
B
If three or more lines intersect at a point, we say that
they are concurrent at that point.
If three or more points are on the same line, we say that
they are collinear.
The Plane A surface is that which has length and
breadth. When a surface is such that the line joining
any two arbitrary points in it lies wholly in the surface,
it is called a plane.
A line in a plane divides the plane in two half-planes.
The Angle In a plane, consider two half-planes bounded
by two lines concurrent at the point O. The intersection
of the two half-planes is an angle. Alternatively, The
inclination of two right lines extending out from one
point in different directions is called a rectilineal angle.
The two lines are called the legs, and the point the
vertex of the angle.
A particular angle in a figure is denoted by three letters,
as BAC, of which the middle one, A, is at the vertex,
and the other two along the legs. The angle is then
read BAC.
b
b
B
Aα
b
C
The angle formed by joining two or more angles together
is called their sum. Thus the sum of the two angles
ABC, P QR is the angle formed by applying the side QP
to the side BC, so that the vertex Q shall fall on the
vertex B, and the side QR on the opposite side of BC
from BA.
When the sum of two angles BAC, CAD is such that
the legs BA, AD form one straight line, they are called
supplements of each other.
When one line stands on another, and makes the adjacent angles at both sides of itself equal, each of the
angles is called a right angle, and the line which stands
on the other is called a perpendicular to it. Hence a
right angle is equal to its supplement.
An acute angle is one which is less than a right angle.
An obtuse angle is one which is greater than a right
angle.
The supplement of an acute angle is obtuse, and conversely, the supplement of an obtuse angle is acute.
When the sum of two angles is a right angle, each is
called the complement of the other.
The Circle
A circle is a plane figure formed by a curve called the
circumference, and is such that all segments drawn
from a certain point within the figure to the circumference are equal to one another. This point is called the
centre.
A radius of a circle is any right line drawn from the
centre to the circumference.
A diameter of a circle is a right line drawn through the
centre and terminated both ways by the circumference.
Four or more points found on the same circle are called
conciclic.
The Polygon
A figure bounded by three or more segments is usually
called a polygon. The segments are called the sides of
the polygon.
A polygon of three sides is called a triangle.
A polygon of four sides is called a quadrilateral.
A polygon which has five sides is called a pentagon; one
which has six sides, a hexagon, and so on.
The Triangle
A triangle whose three sides are unequal is said to be
scalene; a triangle having two sides equal, to be isosceles; and and having all its sides equal, to be equilateral.
A right-angled triangle is one that has one of its angles
a right angle. The side which subtends the right angle
is called the hypotenuse.
An obtuse-angled triangle is one that has one of its
angles obtuse.
An acute-angled triangle is one that has its three angles acute.
An exterior angle of a triangle is one that is formed by
any side and the continuation of another side.
Hence a triangle has six exterior angles; and also each
exterior angle is the supplement of the adjacent interior
angle.
Parallel lines are straight-lines which, being in the same
plane, and being continued to infinity in each direction,
meet with one another in neither (of these directions).
Basic Axioms of Euclidean geometry
Euclid split his set of axioms of plane geometry into 5
postulates and 5 common notions of plane geometry.
These were as follows:
P(1) A straight line segment can be drawn joining any
two points.
P(2) Any straight line segment is contained in a unique
straight line.
P(3) Given any straight line segment, a circle can be
drawn having the segment as radius and one endpoint as center.
P(4) All right angles are congruent.
P(5) If a straight line intersecting two other straight lines
makes the sum of the interior angles on the same
side of itself less than two right angles, the straight
lines, if continued indefinitely, will meet on that side
on which the sum of the angles is less than two right
angles.
CN(1) Things which are equal to the same thing are also
equal to one another.
CN(2) If equals be added to equals, the wholes are equal.
CN(3) If equals be subtracted from equals, the remainders
are equal.
CN(4) Things which coincide with one another are equal
to one another.
CN(5) The whole is greater than the part.
Common notion (2) means: If a1 = a2 and b1 = b2 then
a1 + b1 = a2 + b2. This is true for numbers as well as for
segments and angles.
Common notion (3) means: If a1 = a2 and b1 = b2 then
a1 − b 1 = a2 − b 2 .
Common notion (4) means: If we can move a figure
(angle or segment) to fit exactly on top of the other,
then it means they are equal (in terms of size).
Common notion (5) means: This means that we can
show an object to be smaller than another object by
moving the smaller object until it fits inside the larger
one, thus becoming a part of it.
Note: Postulate (2) is different from the statement that
”Any line contains a segment”, although this statement
is true as well, and it is part of our basic intuition of lines
as being made by points and containing the segments
between their points. Formally though, the difference
between Postulate (2) and the statement above is like
the one between ”for any job, there is a man/woman
who can do it” and ”for any man/woman who can do
a job, there is a job” (sadly, the last statement being
false at this time, even if the first might be true).
In truth, Euclid’s axioms are not enough in themselves
for formally deducing the wealth of beautiful theorems
of the Euclidean geometry, or even for defining notions
like ”equal things”, or for comparing angles and segments. Apart from the axioms, Euclid also relied on
other ”common sense”, intuitive notions like boundedness, in-between-ness, rigid motion, uniqueness,
and basic notions of topology of the plane.
Rigid motion The notion of rigid motion is necessary
when comparing geometric objects. A rigid movement
of a geometric figure in plane can be understood as
cutting the figure out of a sheet of paper representing
the plane and placing it in a different position, except
that the sheet of paper is supposed to be boundless.
In practice, we don’t cut figures out in order to move
them – we clone them! (copy them exactly) by means
of markings on a ruler (for segments) or protractor (for
angles). Moving segments and angles is then enough to
move every other plane figure, no matter how complex.
As a basic tenet of Euclidean geometry, you can
thus move any geometric figure found somewhere
in the plane to any other position in plane. Interestingly, Euler had put quite some effort into proving
this tenet for movements of segments, while taking it
for granted in the case of angles.
Rigid motion by means of a ruler and protractor is so
ingrained in our way of doing geometry that we don’t
even notice how the notion of measure (centimeters,
meters, inches etc. for segments, and degrees for angles) is in fact an indirect process resulting from being
able to compare and add objects by moving them in
suitable places.
Two geometric figures D and D′ are then called equal
or congruent if one can move the first figure and superpose it exactly on top of the second figure, such that
the points of the two figures now coincide. In this case
we write D ≡ D′ .
A segment AB can be said to be smaller than another
one CD if one can move the segment AB until A coincides with D and B is in between C and D. Similarly,
an angle AOB can be said to be smaller than CO′ D if
one can move AOB such that O falls over O′ , the line
OA over O′ C ′ , and B is in the interior of the angle CO′ D.
Angle measures We can define 1◦ as the measure of an
angle such that the sum of 90 angles equal to it makes
a right angle.
This makes sense due to the definition of a sum of two
angles given above and Euclid’s Postulate (4). Euclid
doesn’t tell us that such an angle exists! But we assume
it anyway.
Note: Degrees are defined based on the notion of right
angles (and the assumption that they are all equal), so
if you try to define a right angle as being 90◦, you’d
be moving in circles... Similarly if you tried to define
supplements as summing up to 180◦.
Other basic assumptions that Euler forgot to state
Suppose one would want to describe Euclidean geometry in such a way that even a blind alien would be able
to understand it. One could trail through the Elements
to find all the basic assumptions that Euler has forgotten to state. Apart from the one mentioned above, we
could find:
Basic uniqueness statements. The line and circle
postulated in Postulates (1) and (3), respectively, are
unique.
A bounded figure is called convex if, for any two points
A and B in the interior of the figure, the segment AB
is also in the interior of the figure.
Basic notion of topology. If a line contains a point
found in the interior of a convex figure, then it intersects
that figure in exactly two points.
However, even after gathering all these basic assumptions in a set of Axioms, there would be some work to
be done. One would have to eliminate the superfluous
assumptions, i.e. those which can be considered as theorems or propositions based on the other axioms. For
example, we do not need to assume rigid motion for
all figures - only for angles and segments. On the other
hand, Euclid proved that a segment can be moved to any
other position if we assume that two circles, each passing through the interior of the other, intersect. Another
problem may appear if some of the axioms introduced
actually contradict other axioms. To prove that the axioms are not contradictory, one would have to construct
a model of the plane for which all the axioms hold true,
using other known mathematical objects like numbers,
vector spaces, etc.
Towards the end of the 19th century, David Hilbert initiated an immense effort of constructing a sound axiomatic basis for each area of mathematics. His lectures
at the university of Göttingen in 1898–1899, published
under the title Foundations of Geometry, proposed a
larger set of axioms substituting the traditional axioms
of Euclid. Hilbert proved that his axioms are independent and non-contradictory (relying on algebra and coordinates to construct a model of the plane satisfying his
axioms). Since then, the algebraic/analytic approach to
Euclidian geometry has become dominant. Time permitting, we will discuss Hilbert’s approach towards the
end of the course.
Independently and contemporaneously, a 19-year-old American student named Robert Lee Moore published an
equivalent set of axioms. Some of the axioms coincide, while some of the axioms in Moore’s system are
theorems in Hilbert’s and vice-versa.