文档介绍：Chapter 2 4
Synthetic Geometry
Synthetic geometry is that kind of geometry which deals purely with geometric
objects directly endowed with geometrical properties by abstract axioms. This is in
contrast with a procedure which constructs geometric objects from other things; as,
for example, analytic geometry which, with the artifice of a coordinate system,
models points by n-tuples of numbers. Synthetic geometry is the kind of geometry
for which Euclid is famous and that we all learned in high school.
Modern synthetic geometry, however, has a more plete and consistent
foundation. In this chapter the pattern of this foundation will be adapted, informed
by the previous physical considerations, to develop a synthetic system of axioms
which do not entail such things as uniformity or isotropy. This geometry is a global
one but it is hoped that its elaboration, like that of Euclidean geometry, will be
instructive for the development of a similar, but more general, local theory.
Incidence Geometry
The most basic of geometrical notions are introduced by the concept of an incidence
geometry. Assuming basic ideas from set theory, it is abstractly defined in terms of a
set P whose elements will be called points and a collection of subsets L of P called
lines satisfying three axioms.
Axiom 1: Every line contains at least two points.
Axiom 2: There are three points which are not all contained in one line.
Axiom 3: There is a unique line containing any two points.
The points referred to in these axioms are, of course, all distinct. Generally, points
will be denoted by capital letters from beginning or middle of the alphabet and lines
will be denoted by small letters from the middle of the alphabet.
Chapter 2 5
The first axiom just rules out vacuity. The second axiom guarantees at least two
dimensions; otherwise it would hardly be worth calling a geometry. The third
 
incidence axiom has more substantial content. It both requires that th