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...
