Lecture Notes 6

Non-Euclidean Geometry is not not Euclidean Geometry. The term is usually applied only to the special geometries that are obtained by negating the parallel postulate but keeping the other axioms of Euclidean Geometry.

History of the Parallel Postulate.

Saccheri (1667-1733) "Euclid Freed of Every Flaw" (1733, published posthumously)
The first serious attempt to prove Euclid's parallel postulate by contradiction. This Jesuit priest succeeded in proving a number of interesting results in hyperbolic geometry, but reached a flawed conclusion at the end of the work.
Lambert (1728 - 1777) "Theory of Parallels" (also published posthumously)
Similar in nature to Saccheri's work, and probably influenced by it. However, Lambert was astute enough to realize that he had not proved the parallel postulate. He did not publish this work himself.
Nikolai Ivanovich Lobachewsky (1793-1856) "On the Principles of Geometry" (1829)
The first published account of hyperbolic geometry, in Russian. Lobachewsky developed his ideas from an analytical (trigonometric) viewpoint.
Johann ('Janos') Bolyai (1802-1860) "Appendix exhibiting the absolute science of space: independent of the truth or falsity of Euclid's Axiom XI (by no means previously decided)" in Wolfgang Bolyai's, Essay for studious youths on the elements of mathematics (1832)
An approach similar to Lobachewsky's, but he was unaware of Lobachewsky's work.
Gauss (1777-1855) - unpublished.
In a letter to Wolfgang Bolyai, after receiving a copy of Janos' appendix, Gauss wrote:
If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. So I remained quite stupefied. So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime. ... It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend, who takes the precedence of me in such a remarkable manner.
Later, in a letter to Bessel in 1829, Gauss wrote:
It may take very long before I make public my investigations on this issue: in fact, this may not happen in my lifetime for I fear the "clamor of the Boeotians."
Boeotia was a province of ancient Greece whose inhabitants were known for their dullness and ignorance.
Riemann (1826-1866) "On the Hypotheses Which Lie at the Foundation of Geometry" (1854)
In this inaugural address Riemann outlined the basic ideas underlying Elliptic Geometry

Hyperbolic Geometry

Characteristic Postulate
Through a given point C, not on a given line AB, passes more than one line in the plane not intersecting the given line.

Theorem 9.1: Through a given point C, not on a given line AB, pass an infinite number of lines not intersecting the given line.

Def: Given a point C not on a line AB, the first line through C in either direction that does not meet AB is called a parallel line. Other lines through C which do not meet AB are called nonintersecting lines. The two parallel lines through C are called the right-hand parallel and left-hand parallel. The angle determined by the line from C perpendicular to AB and either the right or left hand parallel is called the angle of parallelism.

Note: While it is true that there is a first line through C which does not meet AB, there is no last line through C which does meet AB.

Theorem 9.2: The two angles of parallelism for the same distance are congruent and acute.

Def: All the lines that are parallel to a given line in the same direction are said to intersect in an omega point (ideal point).

Def: The three sided figure formed by two parallel lines and a line segment meeting both is called an Omega triangle.

Theorem 9.3: The axiom of Pasch holds for an omega triangle, whether the line enters at a vertex or at a point not a vertex.

Theorem 9.4: For any omega triangle AB, the measures of the exterior angles formed by extending AB are greater than the measures of their opposite interior angles.

Theorem 9.5: Omega triangles AB and A'B'' are congruent if the sides of finite length are congruent and if a pair of corresponding angles at A and A' or B and B' are congruent.

Theorem 9.6: Omega triangles AB and A'B'' are congruent if the pair of angles at A and A' are congruent and the pair of angles at B and B' are congruent.

Def: Saccheri quadrilateral: A quadrilateral with two opposite sides of equal length, both perpendicular to a third side. The side of the quadrilateral which makes right angles with both the equal length sides is called the base, and the fourth side is called the summit.

Theorem 9.7: The segment joining the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both.

Theorem 9.8: The summit angles of a Saccheri quadrilateral are congruent and acute.

Def: Lambert quadrilateral: A quadrilateral in which three of the angles are right angles.

Theorem 9.9: The fourth angle of a Lambert quadrilateral is acute.

Theorem 9.10: The sum of the measures of the angles of a right triangle is less than .

Theorem 9.11: The sum of the measures of the angles of any triangle is less than .

Def: Defect: The defect of a triangle is the difference between and the sum of the angles of the triangle. Thus, Theorem 9.11 says that in hyperbolic geometry, every triangle has positive defect. In Euclidean geometry, every triangle has defect zero.

Theorem 9.12: The sum of the measures of the angles of any convex quadrilateral is less than 2.

Theorem 9.13: Two triangles are congruent if the three pairs of corresponding angles are congruent.

Def: Two non-intersecting lines are said to meet at a gamma point (ultra-ideal point).

Theorem 9.14: Two non-intersecting lines have a common perpendicular.

Def: Two polygons are called equivalent if they can be partitioned into the same finite number of pairs of congruent triangles.

Theorem 9.15: Two triangles are equivalent iff they have the same defect.

Def: The area of a triangle is A = kd, where d is the defect and k is a positive constant that is the same for all triangles. (Equivalent polygons have the same area).

Curves in Hyperbolic Geometry

Def: Two points, one on each of two lines, are called corresponding points if the two lines form congruent angles on the same side with the segment whose endpoints are the two given points. (This definition is valid if the two lines meet at an ordinary point, an omega point or a gamma point)

Def: A limiting curve (horocycle) is the set of all points corresponding to a given point on a pencil of rays with an ideal point as vertex.

Def: An equidistant curve is the set of all points corresponding to a given point on a pencil of rays with a common perpendicular.

Theorem 9.16: Three distinct points on a limiting curve uniquely determine it.

Theorem 9.17: Any two different limiting curves are congruent.

Elliptic Geometry

Characteristic Postulate of Elliptic Geometry
Any two lines in a plane meet at an ordinary point.

Other modifications of Euclidean axioms are needed to get a consistent set of axioms for this geometry. These include:

  1. Lines are boundless rather than infinite.
  2. Circles do not have to always exist.
  3. Some pairs of points may be joined by an infinite number of lines.

Theorem 9.18: The segment joining the midpoint of the base and summit of a Saccheri quadrilateral is perpendicular to both the base and summit.

Theorem 9.19: The summit angles of a Saccheri quadrilateral are congruent and obtuse.

Theorem 9.20: A Lambert quadrilateral has its fourth angle obtuse, and each side of this angle is shorter than the opposite side.

Theorem 9.21: The sum of the measures of the angles of any triangle is greater than .

Theorem 9.22: The sum of the measures of the angles of any quadrilateral is greater than 2.


Recall that an axiom system is said to be consistent if no logical contradictions can be derived from these axioms. Proving that a system is consistent is frequently too difficult to be done. We are often satisfied if we can show that one axiom system is "just as" consistent as another, well known system, even if we can't prove that either are absolutely consistent.

Def: Two axiom systems are said to be relatively consistent if any contradiction derived in one of the systems implies that there is a contradiction that can be derived in the other.

The relative consistency of axiom systems can be proved by providing a model of one system inside the other. In terms of the geometries we have studied, if we can model, say, hyperbolic geometry, using only the objects and relations of Euclidean geometry, then hyperbolic geometry and Euclidean geometry would be relatively consistent since the model would permit translation of any contradiction in one system into a contradiction in the other system. E. Beltrami (1868) is credited with doing this for the first time. Since then several other models have been created. Thus, hyperbolic (and elliptic as well) geometry is relatively consistent with Euclidean geometry. Another way to say this is that all three geometries are equally logically valid.

Klein (1849-1925) and Poincaré(1854-1912) models for hyperbolic geometry.