Portal:Euclidean geometry/Chapter 1

This chapter is the introduction to geometry, and introduces some basic, essential, concepts of geometry, which include the concepts of undefinable terms, postulates, and theorems. To see the textbook related to this chapter, see the Wikibook chapter.

Points
In Euclid's Elements in Book 1 and Definition 1, Euclid addresses the definition of Point. "The Elements is the prime example of an axiomatic system from the ancient world. Its form has shaped centuries of mathematics. An axiomatic system should begin with a list of the terms that it will use. This definition says that one term that will be used is that of point." As stated by David E. Joyce Professor of Mathematics and Computer Science at Clark University.

Euclid writes: A point is that which has no part

This is a difficult concept to grasp to be sure but a definition is needed and this is what the father of geometry gives us.


 * [[Image:Point A.svg|40 px]]


 * Every point is represented with the dot, and the letter that labels it. All labels are uppercase for points. This is important because the opposite is true for lines.

Lines
Euclid, in Definition 2 states:

A line is breadthless length.

"Line" is the second primitive term in the Elements. The description, "breadthless length," says that a line will have one dimension, length, but it won't have breadth or depth.


 * Although lines have no width, we do draw them that way; it's hard to draw a widthless line!
 * [[Image:Line.PNG]]


 * This line has two parts, like the point. It has a figure that represents the line, and a label for the line, in this case b.  Notice that the label is lowercase. It is key to be able to identify lines from points.
 * To represent the fact that lines go on forever, we have a pair of arrows on the line. These show the way the line continues on forever in.

Euclid, in Definition 3 states:

The ends of a line are points.

This statement can be taken as indicating that between certain lines and points a relation holds, that a point can be an end of a line. It doesn't say what ends are. It also doesn't indicate how many ends a line can have. For instance, the circumference of a circle has no ends, but a finite line has its two end points.

Euclid, in Definition 4 states:

A straight line is a line which lies evenly with the points on itself.

This statement indicates, at least, that the term "straight line" refers to a kind of line. It is hard to tell what else it means, if anything. Various commentators have interpreted in a variety of ways. There are a some postulates that come a little later in Book I and give meaning to straight lines. Postulate 1 says that a straight line can be drawn between any two points.

Axioms/Postulates
The definition of an axiom is that it is a statement which is taken to be self-evident, and cannot be proved.