文档介绍：Chapter 4 41
Two Dimensions
The initial results which have been obtained so far are independent of dimension and
it is dimension 4 which is of physical interest. However, most the rest of this paper
will be specialized to the case of two dimensions by taking Paschs Postulate as a
new axiom. When doing research it is always methodologically prudent to
investigate simple cases because what is learned in solving them is often the key to
the harder ones; this is the stage of the present work.
Paschs Postulate
To state this axiom first requires the formal introduction of the concept of a triangle
which has hitherto been used implicitly. For any three noncollinear points A, B and
C define the triangle ABC = AB È BC È CA . The points are called the vertices and
the segments are called the sides of the triangle. Note that it is evident from this
definition that the order in which the defining vertices are specified is irrelevant.
Axiom 7. If a line intersects a triangle then it intersects two sides of the
triangle, possibly at mon vertex.
Corollary. If A, B and C are three noncollinear points not on a line m then m cannot
intersect all three sides of the triangle ABC.
Proof. Suppose m intersects AB at D, BC at E and CA at F then, without loss of
generality, take E to be between D and F. The line containing BC then intersects the
segment DF of the t