Euclid's Mathematical System

We will accept the Rules of Reasoning that we have been discussing as our logic rules in our mathematical system. There are several ways in which we can go about setting forth our axioms and undefined terms for geometry. We can keep the number of axioms down to a very few, in which case we need a large number of undefined terms, relations, and operations. We can keep the number of undefined terms, relations, and operations to a small number, thus requiring a larger number of axioms. We will actually work with the Axiom system set up by David Hilbert in 1899 in his book Foundations of Geometry. He not only clarified Euclid's definitions but also filled in the gaps in some of Euclid's proofs. Hilbert recognized that Euclid's proof for the side-angle-side criterion of congruence in triangles was based on an unstated assumption (the principle of superposition) and that this criterion had to be treated as an axiom. He built on the earlier work of Moritz Pasch, who in 1882 published the first rigorous treatise on geometry; Pasch made explicit Euclid's unstated assumptions about betweenness. At the same time we could have used the axioms set forth by Garret Birkhoff. These axioms are the ones with which you are more familiar, being the basis for the texts in High School Geometry. I have included both sets of axioms, so that you can compare the two. We will work exclusively with the Hilbert axioms. An interesting exercise is to prove everything that we will prove using Birkhoff's axioms. Some proofs are easier, and some are harder.

First, let us set our undefined terms. They will be

• point, 
• line, 
• incidence, (a point being incident with a line) 
• betweenness, 
• congruence. 

These objects are to be taken as undefined terms. We will not attempt to make a definition of any of them. However, when we wish to look at a specific example of our geometry, called a model, we will have to establish how each of these undefined terms is to be interpreted. For example, if we wish to look at the model where points are ordered pairs coming from the usual Cartesian plane and lines are the usual straight lines in the Cartesian plane, then we would be able to easily state what it means for a point to be incident with a particular line, what it means for one point to lie between two other points and what it means for two line segments to be congruent. We could change our example though to be a set of seven points, where lines would be ordered pairs of points, incidence would be set inclusion and betweenness and congruence would be meaningless. The particular model that we choose will determine our interpretation of the terms, but will not define them.

At this point we need to make some definitions. In order to do this we need our axiom system. I shall include all of the axioms here, though we will discuss them in more detail later.

Hilbert's Axioms for Neutral Geometry

GROUP I: INCIDENCE AXIOMS 

I-1:

For every point P and for every point Q not equal to P there exists a unique line that passes through P and Q.

I-2:

For every line there exist at least two distinct points incident with .

I-3:

There exist three distinct points with the property that no line is incident with all three of them.

GROUP II: BETWEENESS AXIOMS 

B-1:

If , then A, B, and C are three distinct points all lying on the same line, and .

B-2:

Given any two distinct points B and D, there exist points A, C, and E lying on such that , , and .

B-3:

If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.

B-4:

(PLANE SEPARATION AXIOM) For every line and for any three points A, B, and C not lying on :

(i)

if A and B are on the same side of and B and C are on the same side of , then A and C are on the same side of .

(ii)

if A and B are on opposite sides of and B and C are on opposite sides of , then A and C are on the same side of .

GROUP III: CONGRUENCE AXIOMS 

C-1:

If A and B are distinct points and if A' is any point, then for each ray r emanating from A' there is a unique point B' on r such that and .

C-2:

If and , then . Moreover, every segment is congruent to itself.

C-3:

If , A'*B'*C', , and , then .

C-4:

Given any and given any ray emanating from a point A', then there is a unique ray on a given side of line such that .

C-5:

If and , then . Moreover, every angle is congruent to itself.

C-6:

(SAS) If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.

GROUP IV: CONTINUITY AXIOMS 

ARCHIMEDES' AXIOM:

If AB and CD are any segments, then there is a number n such that if segment CD is laid off n times on the ray emanating from A, then a point E is reached where .  

DEDEKIND'S AXIOM:

Suppose that the set of all points on a line is the union of two nonempty subsets such that no point of is between two points of and vice versa. Then there is a unique point, O, lying on such that if and only if . 

(The following two Principles follow from Dedekind's Axiom, yet are at times more useful.)

CIRCULAR CONTINUITY PRINCIPLE:

If a circle has one point inside and one point outside another circle , then the two circles intersect in two points. 

ELEMENTARY CONTINUITY PRINCIPLE:

If one endpoint of a segment is inside a circle and the other outside the circle, then the segment intersects the circle. 

Birkhoff's Axioms for Neutral Geometry

The setting for these axioms is the Absolute (or Neutral) Plane. It is universal in the sense that all points belong to this plane. It is denoted by .

AXIOM 1:

There exist nonempty subsets of called lines, with the property that each two points belong to exactly one line.

AXIOM 2:

Corresponding to any two points there exists a unique number , the distance from A to B, which is 0 if and only if A = B.

AXIOM 3:

(Birkhoff Ruler Axiom) If k is a line and R denotes the set of real numbers, there exists a one-to-one correspondence () between the points and the numbers such that

where .

AXIOM 4:

For each line k there are exactly two nonempty convex sets satisfying

i)

ii)

. That is, they are pairwise disjoint.

iii)

If then .

AXIOM 5:

To each angle there exists a unique real number x with which is the (degree) measure of the angle

AXIOM 6:

If , then

AXIOM 7:

If is a ray in the edge, k, of an open half plane H(k;P) then there exist a one-to-one correspondence between the open rays in H(k;P) emanating from A and the set of real numbers between 0 and 180 so that if then

AXIOM 8:

(SAS) If a correspondence of two triangles, or a triangle with itself, is such that two sides and the angle between them are respectively congruent to the corresponding two sides and the angle between them, the correspondence is a congruence of triangles.

Definition: The plane  is the collection of all points and lines.

Definition: The segment

A and B are the endpoints of the segment AB.  

Definition: Given points O and A in the plane. The set of points P such that is a circle  with center O. Each segment OP is called a radius .

Definition: The ray 

emanates  from A and is part of .

Definition: The rays and are opposite if they are distinct, emanate from A, and are part of the same line . 

Definition: An angle  with vertex  A is a point A together with 2 non-opposite rays and , called the sides , emanating from A. Denote this angle by .

Definition: If and have a common side and the other two sides and form opposite rays, the angles are supplements or supplementary angles  .

Definition: An angle is a right angle   if it has a supplementary angle to which it is congruent.

Definition: Two lines and m are parallel  if they do not intersect; i.e., if no point lies on both of them. Denote this by .

Definition: Two lines and m are perpendicular   , , if they have a point A in common and there exist rays , a part of , and , a part of m, such that is a right angle.