A  FEEBLE ATTEMPT TO EXPLAIN WHY WE STUDY
MATHEMATICS IN GENERAL, AND GEOMETRY IN PARTICULAR.

In the great periods of culture, which preceded the present one, almost all educated people valued mathematics.  There is little doubt that the growth of mathematics is attributed, at least in part, to the growth of industrialized society and the need for better ways to quantify it.  It is imperative that students in the current and future generations be exposed to as much mathematics as possible.  Even though much of modern mathematics is not easily comprehended the essential nature and accomplishments of the subject can still be understood by anyone willing to give some effort toward learning it.

Before our modern society became so caught up in the fast pace of life we currently live students were not exposed to mathematics until they were somewhat older.  Often, not getting any formal mathematics instruction until they had passed their tenth birthday.   In our current rush to get students through school at the earliest possible age we have come to require that they learn about numbers and operations with numbers while many of them are still too young to fully appreciate them.  It is therefor incumbent upon students to realize this fact and to try and compensate for it by making every effort to understand all the mathematics that is being taught.  Mathematics warrants attention not only because it is a required subject in schools, but because it contains the weighty and beautiful ideas which lend themselves to powerful applications, and because understanding and appreciating mathematics gives us the reasoning skills to better understand our world and how we fit into it.

Mathematics is concerned primarily with what can be accomplished by reasoning.  Those who are opposed to learning how to reason can readily supply evidence that suggests that reasoning is a dispensable activity.  They contend that people can always get by using only the information gleaned from the senses.  However, that information is often limited, and it must always be interpreted, thus making it subjective and fallible.

Mathematics is ideally suited to prepare the mind for higher forms of thought because not only does it pertain to the world of physical things it also deals with abstract concepts.  Through the study of mathematics we learn to pass from concrete forms to abstract forms; this study allows us to move away from the contemplation of the sensible and perishable toward the eternal ideas.  These are the abstractions that are on the same mental level as the concepts of mathematics.  According to Socrates (circa 469 – 399 B.C. ) “The understanding of mathematics is necessary for a sound grasp of ethics.”

We need to understand what mathematics is, how it functions, what it accomplishes for the world, and what it has to offer in itself.  We shall try to see that mathematics is an integral part of our modern world, one of the strongest forces shaping its thoughts and actions, a body of knowledge that is valuable to every branch of our culture.  We also need to be aware of the fact that mathematics as a body of reasoning from axioms stems from one source, the Greeks.

Geometry: (from the Greek words Geo = Earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships.  The earliest beginnings of geometry may be traced to Ancient Egypt.  The Rhind Mathematical Papyrus describes an astoundingly precise means of obtaining an approximation of Pi, accurate to within less than one per cent.

From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry.  Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.

Because of its immediate practical applications, geometry was one of the first branches of mathematics to be developed.  The Egyptians and the Romans developed useable, practical mathematics, but it was the Greeks who took mathematics to the limit suggesting that only the abstract mathematical ideas had any merit.  They argued that conclusions drawn from analogy, or induction, have only a probability of being correct while conclusions drawn from deduction necessarily must be true.  Geometry was the first field to be put on an axiomatic basis, by Euclid.  The Greeks were also interested in many questions about ruler-and-compass constructions.  The next most significant development had to wait until a millennium later, and that was analytic geometry, in which coordinate systems are introduced and points are represented as ordered pairs or triples of numbers.  This sort of representation has since then allowed us to construct new geometries other than the standard Euclidean version.

The central notion in geometry is that of congruence.  In Euclidean geometry, two figures are said to be congruent if they are related by a series of reflections, rotations, and translations.

Although there are many different geometries, we will be studying Euclidian geometry, including, but not necessarily limited to, plane/coordinate geometry, informal geometry, formal geometry, analytic geometry, transformational geometry, and solid geometry.

Formal Geometry:
Suppose you need to solve a crime mystery. You survey the crime scene, gather the facts, and write them down in your memo pad. To solve the crime, you take the known facts and, step by step, show who committed the crime. You conscientiously provide supporting evidence for each statement you make.  Here are the five steps that generally take you through the process:
1. Get or create the statement of the theorem.
2. State the given.
3. Get or create a drawing that represents the given.
4. State what you're going to prove.
5. Provide the proof itself.
This is the application of formal proof.  The formal proof merely justifies/validates the conclusion already made by intuition.  The value of the deductive organization of the proof is that it enables its creator and the reader to test the arguments by the standards of exact reasoning.  The result is a conclusion that is irrefutable.

Euclidean Geometry and Informal Geometry:
Geometry provides students with a way to link their perceptions of the world with the mathematics that allow them to solve a variety of problems they will encounter not only in other disciplines but also in their lives.  Geometry gives students a visual way to conceptualize or organize certain aspects of their environment, whereas algebra provides the tools for dealing with the quantitative aspects of their environment.

Analytic/Coordinate Geometry:
This is the study of the geometry of figures by algebraic representation and manipulation of equations describing their positions, configurations, and separations.  Analytic geometry is also called coordinate geometry.

Transformational Geometry:
The study of translations, rotations, and reflections are the building blocks of transformational geometry.

Solid Geometry:
This is the study of three-dimensional objects.

Why study Geometry?
Over the past 2500 years geometry has been studied because it has been held to be the most exquisite, perfect truth available outside of divine revelation.  It is the surest, clearest way of thinking available to us.  In some ways studying geometry reveals the deepest true essence of the physical world (measuring the earth.)  In addition, studying geometry trains the mind in clear and rigorous ways of thinking (inductive and deductive reasoning).