Chapter One
Some Historical Backgrounds
Chapter One
Some Historical Backgrounds
1-1 Introduction
"When will we ever use this?" This is a question that every teacher has
heard at some point or at several points in time. But a better question would be,
"Where has this been used this in the past?" It is important to not only look to the
future, but to also look to the past. To fully understand a topic, whether it deals
with science, social studies, or mathematics, its history should be explored.
Specifically, to fully understand geometric constructions the history is definitely
important to learn. As the world progresses and evolves so too does geometry. In
high school classrooms today the role of geometry constructions has dramatically
changed.
In order to understand the role of geometry today, the history of geometry
must be discussed. As Marshall and Rich state in the article, The Role of History
in a Mathematics Class.
"... history has a vital role to play in today's mathematics classrooms. It allows
students and teachers to think and talk about mathematics in meaningful ways. It
demythologizes mathematics by showing that it is the creation of human beings.
History enriches the mathematics curriculum. It deepens the values and broadens
the knowledge that students construct in mathematics class."
This quote truly sums up the importance of relating the past to the present.
Students will benefit from knowing about how mathematical topics arose and why
they are still important today.
To thoroughly examine the history of geometry, we must go back to ancient
Egyptian mathematics. A topic that often amazes people is the beautiful geometry
in Egyptian pyramids. The mathematics and specifically geometry involved in the
building of these pyramids is extensive. From Egypt, Thales brought geometric
ideas and introduced them to Greece. This led to the important evolution of Greek
deductive proofs.
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
1
Chapter One
Some Historical Backgrounds
Thales is known to have come up with five theorems in geometry.
1. A circle is bisected by any diameter.
2. The base angles of an isosceles triangle are equal.
3. The vertical angles between two intersecting straight lines are equal.
4. Two triangles are congruent if they have two angles and one side equal.
5. An angle in a semicircle is a right angle.
However, the title of the "father of geometry" is often given to Euclid.
Living around the time of 300 BC, he is most known for his book The Elements.
He took the ideas of Thales and other mathematicians and put them down in an
organized collection of definitions, axioms and postulates. From these basics, the
rest of geometry evolves. In The Elements, the first four definitions are as follows:
1. A point is that which has no part.
2. A line is breadthless length.
3. The extremities of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
The basic five axioms (postulate) given by Euclid's in his famous book are the
following axioms:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That if a straight line falling on two straight lines makes the interior
angles on the same side less than two right angles, the straight lines, if
produced indefinitely, will meet on that side on which the angles are
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
2
Chapter One
Some Historical Backgrounds
less that two right angles.
The first two axioms known as Euclidean Straightedge, (a kind of
idealized ruler that is unmarked but indefinitely extendible) and the thirst axiom
is known as Euclidean Compass (theoretical ruler and compass). and a Compass
(two arms connected by a hinge, with a sharp spike on the end of one arm and a
drawing implement on the end of the other).
Sir Thomas Heath wrote a respected translation of Euclid's The Elements in
1926 entitled The Thirteen Books of Euclid's Elements. This translation seems to
be the most accepted version of Euclid's writings given modifications and
additions. Since the time of Euclid there have been three famous problems which
have captivated the minds and of many mathematicians. These three problems of
antiquity are as follows:
1.
Squaring the Circle
2.
Doubling the Cube
3.
Trisecting an Angle.
Far back in history and to this present day, these problems are is cussed in
detail. In early geometry, the tools of the trade were a compass and straightedge.
A compass was strictly used to make circles of a given radius. Greeks used
collapsible compasses, which would automatically collapse. Nowadays, we use
rigid compasses, which can hold a certain radius, but is has been shown that
construction with rigid compass and straightedge is equivalent to construction
with collapsible compass and straightedge. However, compasses have changed
dramatically over the years. Some compasses have markings used to construct
circles with a given radius. Of course, under the strict rules of Greeks, these
compasses would not have been allowed.
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
3
Chapter One
Some Historical Backgrounds
Figure1.1
Construction process is one of the main subjects in Euclid's geometry by drawing
with Euclidean Straightedge and Compass. More strictly, there were no markings
on the straightedge. A straightedge was to be used only for drawing a segment
between two points. There were very specific rules about what could and could
not be used for mathematical drawings. These drawings, known as constructions,
had to be exact. If the rules were broken, the mathematics involved in the
constructions was often disregarded. When describing these concepts to students
nowadays, showing pictures of ancient paintings with these tools help illustrate
the importance and commonplace of geometry and these aforementioned tools.
Figure 1.2 A portion of Raphaello Sanzio's painting The School of Athens from 16th century
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
4
Chapter One
Some Historical Backgrounds
Figure 1.3 The Measurers: A Flemish Image of Mathematics in the 16th century
In regards to the history of constructions, a Danish geometer, Georg Mohr,
proved that any construction that could be created by using a compass and
straightedge could in fact be created by a compass alone. This surprising fact
published in 1672 is normally credited to the Italian mathematician, Lorenzo
Mascherone from the eighteenth century. Hence, constructions created using
only compasses are called Mascheroni constructions.
After Euclid, geometry continued to evolve led by Archimedes, Apollonius
and others. However, the next mathematicians to make a dramatic shift in the
nature of geometry were the French mathematicians, Rene Descartes and Pierre
de Fermat, in the seventeenth century, who introduced coordinate geometry. This
advance of connecting algebra to geometry directly led to other great advances in
many areas of mathematics.
Non-Euclidean geometry was the next major movement. Janos Bolyai,
following the footsteps of his father, attempted to create a new axiom to replace
Euclid's fifth axiom. Around 1824, this study led to development of a new
geometry called non-Euclidean geometry. Another mathematician that made
contributions to the formation of non-Euclidean geometry was Nikolai Ivanovich
Lobachevsky. In 1840, Lobachevsky published Geometrie imaginaire. Because
of Bolyai and Lobachevsky's direct connection to Gauss, some believe that nonEuclidean geometry should in fact be credited to Gauss.
Even now, geometry continues to progress. In addition, how schools
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
5
Chapter One
Some Historical Backgrounds
teach geometry has continued to change. In the past, compass and straightedge
constructions were a part of the curriculum. However, in most recent years,
constructions have faded out. In older textbooks, constructions were entire.
1-2 Geometric Algebra
The early Greeks were able to represent numbers or values by a length, but
they did not have algebraic notation. They were, however, able to represent
algebraic identities and operations geometrically.
1-2-1 Identities
The Greeks were able to establish algebraic identities through the use of
rectangles. Here is an example
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
6
Chapter One
Some Historical Backgrounds
Figure 1.4
Exercise 1: Establish each of the following identities with the use of rectangles.
1.
2.
3.
4.
(a - b)2 = a2 - 2ab + b2
a(b + c) = ab + ac
(a + b)(c + d) = ac + bc + ad + bd
a2 - b2 = (a + b) (a - b)
1-2-2 Operations
The Greeks were also able to construct mathematical operations using
lengths instead of numbers. From given lengths they constructed new lengths that,
had they been numbers, would have been the actual solution. It is possible to do
this for all operations.
To get started, it is necessary to choose a line segment as the unit of
length, 1, which has the property that 1a = a for any length a.
i) Addition(Sum)
Here is a look at how to construct a + b, given a length, a, and a length, b.
Figure 1.5
Using the straightedge, draw a segment. Then, using the compass, measure length
a and mark it off on the segment. Now, measure length b and mark it on the
segment from one of the endpoints of a. This is (a + b).
Exercise 2: Establish some arbitrary lengths, a and b.
1. Construct (a - b).
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
7
Chapter One
Some Historical Backgrounds
2. Construct (2b - a).
Not only can one add and subtract line segments; one can also multiply and
divide them. The product ab and quotient a/b of line segments a and b are
obtained by the straightedge and compass constructions below. The key
ingredients are parallels, and the key geometric property involved is the Thales
theorem on the proportionality of line segments cut off by parallel lines.
ii) Multiplication
Multiplication is another operation that is constructive. This process relies on
the use of proportions.
Figure 1.6
Method I
Construction
1)
Establish a unit segment
2) Establish segment of length a.
3) Establish segment of length b.
4) Construct an x - y axis and label the origin as A.
5) Mark off the unit segment with the compass on the x-axis and label the endpoint B.
6) On the y-axis , mark off the length of segment a, beginning at A, name the endpoint C.
7) From B, mark off the length of segment b, and name the endpoint D.
8) Connect B and C with a segment.
9) From D, construct a line parallel to BC, call the intersection with the y-axis E.
10) The segment CE is the length of ab.
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
8
Chapter One
Some Historical Backgrounds
Proof
Let
Since
Then
Thus
CE = x
BC || DE
x/b = a/1
x = ab
Figure 1.7
Method II
To multiply line segment b by line segment a, we first construct any
triangle UOA with |OU| = 1 and |OA| = a. We then extend OU by length b to B1
and construct the parallel to UA through B1. Suppose this parallel meets the
extension of OA at C (Figure 1.8)
Figure 1.8: the product of line segments
iii) Division
Dividing a line segment into n equal parts.
Given a line segment AB, draw any other line L through A and mark n
successive, equally spaced points Ai,A2,A3,...,An along L using the compass set to
any fixed radius. Figure 1.9 shows the case n = 5. Then connect An to B, and draw
the parallels to BAn through Ai, A2, ... , An-i. These parallels divide AB into n equal
parts
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
9
Chapter One
Some Historical Backgrounds
Figure 1.9: Dividing a line segment into equal parts
This construction depends on a property of parallel lines sometimes
attributed to Thales (Greek mathematician from around 600
BCE):
parallels cut
any lines they cross in proportional segments. The most commonly used instance
of this theorem is shown in Figure 1.10, where a parallel to one side of a triangle
cuts the other two sides proportionally.
The line  parallel to the side BC cuts side AB into the segments AP and
PB, side AC into AQ and QC, and |AP|/ |PB| = |AQ|/| QC|.
Figure 1.10: The Thales theorem in a triangle
This theorem of Thales is the key to using algebra in geometry. Previously,
we see how it used to multiply and in the next section we use it to divide line
segments, and later we investigate how it may be derived from fundamental
geometric principles.
Quotient of line segments
To divide line segment b by line segment a, we begin with the same triangle
UOA with |OU| = 1 and |OA|= a. Then we extend OA by distance b to B2 and
construct the parallel to UA through B2. Suppose that this parallel meets the
extension of OU at D (Figure 1.11).
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
10
Chapter One
Some Historical Backgrounds
Figure 1.11: The quotient of line segments
The sum operation allows us to construct a segment n units in length, for
any natural number n, simply by adding the segment 1 to itself n times. The
quotient operation then allows us to construct a segment of length m/n, for any
natural numbers m and n ≠0. These are what we call the rational lengths. A great
discovery of the Pythagoreans was that some lengths are not rational, and that
some of these "irrational" lengths can be constructed by straightedge and
compass. It is not known how the Pythagoreans made this discovery, but it has a
connection with the Thales theorem, as we will see in the next section.
1-2-3 Greatest Common Factor
We have seen cases in which only two types of tiles are given. Read the example
below and add an example of the Greatest Common Factor to your tool kit. Then
use a generic rectangle to find the factors of each of the polynomials below. In
other words, find the dimensions of each rectangle with the given area.
For 2x2 + 10x, the quantity "2x" is called The Greatest Common Factor.
Although the diagram could have dimensions 2(x2 + 5x), x(2x + 10), or
x(x + 5), we usually choose 2x(x + 5) because the 2x is the largest factor that is
common to both 2x2 and 10x. Unless directed otherwise, when told to factor, you
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
11
Chapter One
Some Historical Backgrounds
should always find the greatest common factor, then examine the parentheses to
see if any further factoring is possible.
Exercise 3: Justify the truth of each of the following expressions.
1. x2 + 7x =[ x(x + 7) ]
2. 3x2 + 6x =[ 3x(x + 2) or 3(x2 + 2x) or x(3x + 6) ]
3. 3x + 6= [ 3(x + 2) ]
Some expressions can be factored more than once. For example we can factor
3x3 - 6x2 - 45x as (3x)(x2 - 2x - 15).
However, x2 - 2x - 15 factors to (x + 3) (x - 5). Thus, the complete factoring of
3x3 - 6x2 - 45x is 3x(x + 3)(x - 5).
Notice that the greatest common factor, 3x, is removed first. Discuss this
example with your study team and record how to determine if a polynomial is
completely factored.
Exercise 4: Factor each of the following polynomials as completely as possible.
Consider these kinds of problems as another example of sub
problems. Always look for the greatest common factor first and write
it as a product with the remaining polynomial. Then continue
factoring the polynomial, if possible.
1. 5x2 + 15x -20
2. x2y - 3xy - 10y
3.
2x2 – 50
Exercise 5: Use the Euclidean Straightedge and Compass to draw each of the
following; In all problems below a segment AB is given by its end points A and B.:
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
12
Chapter One
Some Historical Backgrounds
1. Construct segments 2, 3, 4, etc. times larger than AB.
2. A point C is known to lie outside the straight line AB. Construct a
point D symmetric to C with respect to AB.
3. A circle is given by its radius R and the center O. Assume O does not
lie on AB. Find the points of intersection of the circle with the
segment AB.
4. Find a point C such that AC is perpendicular to AB.
5. Determine whether three given points A, B, C lie on the same line.
6. Given three points A, B and C. C is known to lie outside the straight
line AB. Complete the parallelogram ABCD.
Prepared by Assist Prof. Dr. Ibrahim Othman Hamad, College of Science, Salahaddin University-Erbil. 2018-2019
13