The Parallel Postulate

We have viewed Euclidean geometry as an axiomatic system. In introducing axiomatic systems, we studied three-point geometry, four-line geometry, and Fano's geometry. Recall the two ingredients of an axiomatic system:

• undefined terms
• a set of axioms

We have seen the axioms that are also true in Euclidean geometry as well as the axioms that are invalid in Euclidean geometry.

Now, we are going to study some geometries, other than finite geometries, different from the Euclidean geometry. First, we want to take a closer look at the foundation of Euclidean geometry.

With 23 definitions and several implicit assumptions, Euclid derived much of the (planar) geometry from five postulates (or axioms):

1. A straight line may be drawn between any two points. (Given two distinct points, there is always a line passing through the two points.)
2. A piece of straight line may be extended indefinitely.
3. A circle may be drawn with any given radius and an arbitrary center.
4. All right angles are equal.
5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than two right angles.

Postulates 1 and 3 are based on the geometric construction. Postulate 2 illustrates a common belief that straight lines may not terminate. A right angle was defined in Definition 10 by Euclid as an angle that equals its adjacent angle. So, Postulate 4 is justified by our belief that no matter where two perpendicular lines are drawn, the angle they form is one and the same and is called right.

Compared with the first four postulates, Postulate 5 is much more complicated. It is conjectured that Euclid himself had mixed feelings about the fifth postulate as he avoided using it until Proposition I.29 in his Elements. The fifth postulate, also known as the Parallel Postulate, attracted immediate attention. Many substitutes were suggested and many felt it is a consequence of the other postulates. Now, we know that the fifth postulate is independent of the other postulates and it cannot be derived from the other postulates.

Here's a short list of equivalent statements to the fifth postulate:

1. (Playfair's axiom): Through a point not on a given line, exactly one line can be drawn in the plane parallel to the given line.
2. There exists a pair of similar non-congruent triangles.
3. For any three non-colinear points, there exists a circle passing through them.
4. The sum of the interior angles in a triangle is two right angles.
5. Straight lines parallel to a third line are parallel to each other.
6. There is no upper bound to the area of a triangle.
7. Pythagorean theorem.

Euclid's set of axioms (postulates) is not complete. One example is about betweenness (other examples are contained in our last project, Project III). The following is another example of a statement that is independent and consistent with the five postulates:

Pasch's axiom: A line entering a triangle at a vertex must intersect the opposite side, and a line that intersects one side of a triangle at a point other other a vertex also intersects a second side.

Many modern sets of axioms were devised. We will use the one set up by the great French mathematician David Hilbert in 1899 (100 years ago!) in his book Foundations of Geometry (English edition is still available). Finally, here is the definition of Euclidean geometry (after we talked about it so many times!):

Undefined terms:

• point
• line
• incidence
• betweenness
• congruence

Hilbert's axioms:

• Group I: Axioms of Incidence

  1. Two distinct points always completely determine a straight line.
  2. Any two distinct points of a straight line completely determine that line.
  3. Three points A, B, and C not on the same line always completely determine a plane.
  4. Any three points of a plane, which do not lie on the same line, completely determine that plane.
  5. If two points of a straight line lie in a plane, then every point of the line lies in the plane.
  6. If two planes have a point in common, then they have at least a second point in common.
  7. Upon every line there exists at least two points, in every plane at least three points not lying in the same line, and in space there exist at least four points not lying in a plane.

• Group II: Axioms of Betweenness

  1. If A, B, and C are points of a straight line and B lies between A and C, then B lies also between C and A.
  2. If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D.
  3. Of any three points on a straight line, there is always one and only one which lies between the other two.
  4. Any four points A, B, C, and D of a straight line can always be so arranged that B shall lie between A and C and also between A and D, and furthermore, that C shall lie between A and D and also between B and D.
  5. Let A, B, and C be three points not on the smae line, and let a be a line lying in the plane ABC and not passing through any of the points A, B, and C. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.

• Group III: Axiom of Parallels

  In a plane, there can be drawn through any point A, lying outside of a line, one and only one line that does not intersect the given line.

• Group IV: Axioms of Congruence

  1. If A and B are two points on a line a and if A' is a point on the same or another line a¢, then, upon a given side of A' on the line a¢, we can always find one and only one point B' so that the segment AB (or BA) is congruent to the segment A'B'. We indicate this by writing AB º A'B'.
  2. AB º A'B', and A'B' º A''B'' imply AB º A''B''.
  3. Let AB and BC be two segments of a line that have no points in common aside from the point B, and, furthermore, let A'B' and B'C' be two segments of a line that have no points in common aside from the point B'. Then, if AB º A'B' and BC º B'C', we have AC º A'C'.
  4. (Copy of angles) Let an angle (h,k) be given in the plane a, and let a line a¢ be given in a plane a¢. Suppose also, in the plane a¢, a definite side of the line a¢ be assigned. Denote by h' a half-ray of the straight line a¢ emanating from a point O' of this line. Then, in the plane a¢, there is one and only one half-ray k' such that the angle (h,k) or (k,h) is congruent to the angle (h',k'), and all interior points of the angle (h',k') lie upon the given side of a¢.
  5. angle (h,k) º angle (h',k'), angle (h',k') º angle (h'',k'') imply angle (h,k) º angle (h'',k'').
  6. SAS congruence

• Group V: Axiom of Continuity

  Let A₁ be any point upon a straight line between the arbitrarily chosen points A and B. Take the points A₂, A₃, ..., so that A₁ lie between A and A₂, A₂ lie between A₁ and A₃, ..., and so on. Moreover, let the segments AA₁, A₁A₂, A₂A₃, ..., be equal to one another. Then, among this series of points, there always exists a certain points A_n such that B lies between A and A_n.