**Pappus' Theorem:** If points A,B and C are on one line and A', B' and C' are on another
line then the points of intersection of the lines AC' and CA', AB' and BA', and BC' and
CB' lie on a common line called the Pappus line of the configuration.

- There exists at least one line.
- Every line has exactly three points.
- Not all lines are on the same point. [N.B. Change from the text]
- If a point is not on a given line, then there exists exactly one line on the point that is parallel to the given line.
- If P is a point not on a line, there exists exactly one point P' on the line such that no line joins P and P'.
- With the exception in Axiom 5, if P and Q are distinct points, then exactly one line contains both of them.

*Pf*. Let X be any point. By corrected axiom 3, there is a line not containing X. This line
contains points A,B,C [Axiom 2]. X lies on lines meeting two of these points, say B and C
[Axiom 5]. There is exactly one line through X parallel to BC [Axiom 4]. There can be no
other line through X since by Axiom 4 it would have to meet BC at a point other than A, B
or C [Axioms 6 and 5], and this would contradict Axiom 2.

Pappus geometry has 9 points and 9 lines.

**Desargues' Theorem:** In a projective plane, two triangles are said to be perspective from a point if the three lines
joining corresponding vertices of the triangles meet at a common point called the center. Two triangles are said to be perspective from a line if the three points of intersection of corresponding lines all lie on a common line, called the axis. Desargues' theorem states that two triangles are perspective from a point if and only if they are
perspective from a line.

Desargues' Configuration has 10 points and 10 lines.

*Local Definitions for this geometry only!*

The line l is a *polar* of the point P if there is no line connecting P and a point on l.

The point P is a *pole* of the line l if there is no point common to l and any line on P.

- There exists at least one point.
- Each point has at least one polar.
- Every line has at most one pole.
- Two distinct points are on at most one line.
- Every line has exactly three distinct points on it.
- If a line does not contain a point P, then there is a point on both the line and any polar of P.

**Theorem 1.11** Every line in the geometry of Desargues has exactly one pole.

**Theorem 1.12** Every point in the geometry of Desargues has exactly one polar.

**Def: PG(n,q)**

Correct definition of order in the text.

Talk about ovals.

- Lack of continuity.
- Pasch's Axiom.
- order of points and betweeness
- problems with superposition.

- Pasch
- Hilbert - in the style of Euclid
- Birkhoff - a different approach

- consistent
- complete
- independent