CHAPTER 1
Euclid &4&&5
There are a few mathematicians that have a significant effect on our everyday life. Euclid qualifies on that aspect. He is considered the, "Father," of Geometry and as you look around at the myriad of shapes present in your world that have been created by knowledge of this Geometry, the full effect of that title may start to sink in. The very size of your cell phone, the angle formed by the back of your chair and the seat are all the result of a geometric problem being solved. The word Geometry is Greek for geos, meaning earth and metron, meaning measure.
Let us take a quick stab at the history. When Alexander the Great split his empire among his generals, Egypt was given to a general named Ptolemy Soter who created a library in the Egyptian city he generated and named after Alexander, Alexandria around 332 BC with the mission of collecting all the world's knowledge, no small task.
At this time, the world was going through a great transition in terms of people movement, commerce and influence control. The center of the universe, as far as this dialogue is concerned, was the Mediterranean Sea. Let us look at map of that area of the world in the year of about 300 BC.
We can see that the Roman Empire (white shading) was in its infancy. The Punic Empire (darkest shading) centered in the city Carthage was in direct competition with the developing Roman Empire. History tells us that in another one hundred years the two empires would clash. The highlight of this Punic-Roman war was a Punic general named Hannibal marching over the Alps in what is now northern Italy with elephants. Despite Hannibal's success, the Romans would eventually prevail and the Roman Empire would begin its domination of the Mediterranean area. This would not occur for another one hundred years.
What is interesting is the extent of the Greek influence in the eastern Mediterranean. The success of this influence was to stimulate an exchange of ideas between people. Stability meant that people were beginning to collect in cities causing an urban trend towards trade guilds where people with common skills could work together to pass on their knowledge. The idea of apprentices pledging time to masters so as to learn the necessary skills became the norm. The apprentices became the masters and the cycle continued.
Fortunately our profession took up the same model. Apprentices were students and Masters were Teachers. Society knew that to succeed the passing on and the Fortunately our profession took up the same model. Apprentices were students and Masters were Teachers. Society knew that to succeed the passing on and the subsequent natural increase in knowledge was paramount to survival of the culture. These early "knowledge," guilds were usually based on philosophy as scholars began to look around them at the increasing interactions between people. Some important terms that may be familiar to the education world, came out at this time starting with our word, "University." The Latin word, "universitas," refers to a number of people associated together. Another facet of the school world would be the, "Gymnasium," which comes from Latinization of the Greek phrase, "bodily exercise school where one trains naked, Plato believed that learning best occurred in a healthy body. This was picked up by Hippocrates who swore up and down about it. The mention of Plato brings up his school that he founded in Athens called the Academy. Of particular interest to us is the inscription that appeared of the door to the Academy, "Let None But Geometers Enter Here."
The word, "Library," comes from the Latin word, "liber," that refers to books but specifically the collection of written records. This marks a significant change in the passing on of knowledge. Instead of, "word to mouth," it could be examined by various people at their discretion. The, "mission of collecting all the world's knowledge," taken on by Euclid created the Alexandria Library which at the time was considered the depository of the world's understanding of all things.
The purpose of this narrative is not to examine the individual contributions to our study of geometry. Suffice to say that there was a tremendous exchange of information amongst many philosophers that created a period of growth in the world's knowledge. The difference between philosophers and mathematicians was and is not a great leap. The complexity that these individuals attacked is sometimes quite unbelievable. They worked in what today we would call think tanks but the impression I have been brought up with, is a series of Greek Mathematicians sitting around a sandbox using sticks to draw circles, triangles and other Geometric shapes. I am fairly certain the next, "photograph," is faked.
If one looks carefully at the dates, these Mathematicians probably never met face to face, but their data was shared, Euclid's major contribution to the topic of Geometry was his fifteen books collectively titled the, "Elements." The book is important on two fronts. First it is a collection of our information about all aspects of Geometry. Second it is considered the textbook on how to write a textbook. The flow of stating the salient facts to a topic, explaining those facts, producing solutions to sample problems and then providing problems to allow the reader to practice has been repeated in innumerable manuscripts.
The final historical fact I will touch upon briefly, is the journey of the Elements. The Library of Alexandria disappeared. There are stories of Julius Caesar burning the library, there are stories of it disappearing in a sandstorm. The knowledge contained in the Elements was passed on to the Arabs and eventually returned to Europe in 1120 AD, when the English monk Adelard of Bath translated it into Latin from an Arabic translation. This is a little ironic since Euclid probably originally wrote it in Latin.
The study of Geometry is much different than the study of Algebra. The study of Algebra is the mastery of techniques whereas the study of Geometry is the art of logical reasoning. If one plans to move through the education system in the field of Mathematics, be prepared to see the, "art of logical reasoning," used much more prevalently in the form of a logical proof. Is that not the way of life? You want to influence people by the art of logical persuasion.
&4
This may appear to be a little strange title for a section of work. The purpose of this chapter is to establish Euclidean Geometry, so we need to do some strange things. Any new topic, in any subject, requires that you define your terms. We need to define three undefinable terms. Let us just jump into it!
You should be familiar with a point, let us say Point A
Let us say we bent the paper a little bit and magnified it about 1 000 000X, let us look at the ink spot we have designated Point A
We should be able to see that we are violating our definition. We have an object that has width, length and depth.
This is something that plagues Mathematics and all Sciences for that matter. We assume that a point is just there, but as soon as you designate it, it becomes a thing that has measurements. This will immediately throw off any subsequent measurements adding a degree of uncertainty. Let us continue.
Before we get in to various types of lines, let us use the definition as is which implies a line extending to infinity in each direction. We would call this line [bar.AB]. An explanation of the notation will happen in a moment.
Let us apply the same technique of changing our perspective and increase the magnification.
Once again we assume a line has no width or depth, but the reality is that it does.
We need to be a little more specific on our definition of line. The idea that all lines extend to infinity is admirable, but not always practical. Let us establish some additional terms.
It is important to understand these terms, but in many cases the reader must understand that a problem will only mention line AB. It is rare that the above technicalities will be required.
The plane is where we will begin our study of Geometry. It could be the piece of paper that is in front of you, the desktop, the computer screen the blackboard. Man's use of the plane in various descriptions of reality can be staggering. The 3D TV is simply programmers tricking your eye and brain into seeing three dimensions. Video games that tell you that you are, "right in the action," are using deception to fool you. It is all done on the plane.
We will only be looking at three dimensional Geometry in a later section, but let me end with Euclid's last undefinable definition.
&5
This next section deals with Euclid's Five Postulates. First, a postulate is a piece of knowledge that we all need to agree to so as to proceed with the topic. A synonym for Postulate can be the word axiom. These will NOT be proven but they are logically sound, so they need to be as simple as possible. The reader should notice the addition of the word, "straight."
Euclid's Five Postulates
1) There exists only one straight line between two points.
2) Any straight line segment can be extended indefinitely in either direction.
3) Given any straight line segment, a circle can be drawn using the line segment as the radius and one of the endpoints as the center.
4) All right angles are equal to one another.
5) Given any straight line and a point NOT on the line, there exists one and only one straight line that passes through the point that is parallel to the straight line.
The first four are seemingly quite simple, the fifth a little more complicated. Euclid's Fifth Postulate would become a topic of constant conversation amongst mathematicians. It was the great Gauss who discovered that the fifth Postulate could be ignored and a new type of Geometry could exist. It is named Non-Euclidean Geometry and enters into discussions where straight becomes blurred. I mean straight lines. Einstein showed that a beam of light, something Euclid would see as a perfect straight line, bends from the effect of gravity.
Hopefully, we have established the ground rules. Euclidean Geometry rules much of the world.
"There is no Royal Road to Geometry."
- Euclid -
There is also no Assignment to this chapter
Angle
We need to establish another important facet of Geometry, that of an angle. It requires a little bit of thought, but let us use the following definition.
We note that the diagram at the left we have CLEARLY put the three points A, B and C. Do not expect this detail all the time. We can say we have two line segments with a common point. The common point is called the VERTEX.
We can now define ANGLE ABC. The notation for angle is a little angle - [angle] Hence we would say, [angle] AABC.
Let us try and look at this visually.
We can see the rotation of line AB from line BC. However, what if the line had rotated other way?
We can see this is a much greater rotation. We define the larger rotation or larger angle as a REFLEX angle. Hence if we wanted to denote the situation at the right we would say: reflex [angle] ABC
We now need to create a scale to measure angles. There are three ways of measuring an angle and I am sure you know one of them, but do you know why?
Degree Measurement
We define a complete revolution as 360°. Why did they pick the number 360? The reason goes back to those Greeks. The explosion of information included a lot of observations in Astronomy, the study of the stars, planets and other objects in the sky. The Greeks knew that the earth circled the sun and a year was composed of approximately, 360 days. Greek mathematicians jumped on this fact and said the earth rotated 1° per day. Considering it was about 300BC, they were not far off.
This leads to several elementary conclusions.
Half a complete rotation is a straight line. This STRAIGHT angle is
[MATHEMATICAL EXPRESSION OMITTED]
One quarter of a complete rotation is one line right straight up on the other. This angle is called RIGHT and is
[MATHEMATICAL EXPRESSION OMITTED]
We note above there is a special relationship between line ZW and line WK.
The more common diagram would appear as follows:
[MATHEMATICAL EXPRESSION OMITTED]
For completeness we will look at the two other methods of defining an angle.
Radian Measurement (we will simply introduce this method)
This method of measuring an angle involves a circle with radius one unit(centimeter, kilometer or any unit). If we were considering [angle] ABC, we would put it in the following position.
We notice that in this case each line is one unit and the vertex of the angle is the center of the circle.
Radian Angle Measurement is simply the length of arc AC denoted AC.
From this definition we can see that a complete revolution or 360°, would equal the length of the circumference of the circle. Since C = 2pr and r = 1
[MATHEMATICAL EXPRESSION OMITTED]
From this we can also state 180° = pR and 90° = pR/2. This will be left to a later section.
Gradient Angle Measurement
This form of measurement is used by engineers when they are building roads or train tracks. You may have seen a sign at the side of the road pointing out an approaching hill and stating an increase in, "gradient," of say 10%. First we note that this measurement is a percentage scale that runs from 0% to 100%. Second, let us look at road angles from normal to extreme.
From this we can define Gradient Angle Measurement as the percentage of a 90° angle.
Therefore if a road went up at an angle of 7.2°, we could say that this is gradient of 7.2/90 x 100% = 8% gradient.
Once again we will leave this topic to the engineers.
Five Constructions
Euclid set out five tasks to create geometrical drawings. The topic may be a bit obscure, but it allows people to, "feel," Geometry.
This is definitely, "Old School," and I must admit that I am doing the diagrams on a computer, but have fun with it!
We are going to use terms related to a compass and straight edge. First let us look at a compass.
In these constructions we will never introduce measurements but in later sections it will be necessary that your straight edge to have measurements. Your straight edge can in fact be a ruler.
We will be using terms like, "suitable radius," which simply means not too big, not too small. The other term is bisect which simply means to divide into two equal parts.
Let us take up our tools and get at it.
First we need to define right bisector. The bisector part means to divide the line in half and the right part means to find the line that is at a right angle, or perpendicular. Here we go!
The reader should be aware that this last construction is a little different in the sense that we did not use the phrase, "suitable radius." This would haunt mathematicians for quite some time as it refers to the fifth postulate. As was mentioned before, the great Gauss would create Non-Euclidean Geometry to solve this vexing problem.
The next two constructions were not mentioned by Euclid, but it is certain that he employed them.
Next we will do several questions that line up with the Assignment. These questions should be considered pieces of art, and not your normal tedious Math drudgery.
