Axioms of Classification.—The eight axioms of classification are as follows:
1. Points form a class of entities with at least two members.
2. Any straight line is a class of points containing at least three members.
3. Any two distinct points lie in one and only one straight line.
4. There is at least one straight line which does not contain all the points.
5. If A, B, C are non-collinear points, and A′ is on the straight line BC, and B′ is on the straight line CA, then the straight lines AA′ and BB′ possess a point in common.
Definition.—If A, B, C are any three non-collinear points, the plane ABC is the class of points lying on the straight lines joining A with the various points on the straight line BC.
6. There is at least one plane which does not contain all the points.
7. There exists a plane α, and a point A not incident in α, such that any point lies in some straight line which contains both A and a point in α.
Definition.—Harm. (ABCD) symbolizes the following conjoint statements: (1) that the points A, B, C, D are collinear, and (2) that a quadrilateral can be found with one pair of opposite sides intersecting at A, with the other pair intersecting at C, and with its diagonals passing through B and D respectively. Then B and D are said to be “harmonic conjugates” with respect to A and C.