A SET OF AXIOMS FOR EUCLIDEAN GEOMETRY

An _ABSTRACT GEOMETRY_ consists of a set of points and lines such that

 (i)  given two points A and B, some line  l  contains both;
 (ii) every line contains at least two points

(In this formulation, each line constitutes a set of points, namely the
points it contains, while points have no further structure.  Often one
wishes to put the concepts ``point'' and ``line'' on an equal footing.
In this case one begins with a set of points and a set of lines, and
instead of saying ``line  l  contains point  p'' one says ``line  l  and 
point  p  are incident.'')



We call an abstract geometry  an INCIDENCE GEOMETRY if

 (i)  every two distinct points in P lie on a _unique_ line;
 (ii) there exist three points which do not lie all on one line.


A distance function d assigns to every pair of points  A  and  B  a
non-negative number d(A,B) in such a way that

 (i)   A = B  implies  d(A,B) = 0;
 (ii)  d(A,B) = 0  implies  A = B;
 (iii) d(A,C) never exceeds  d(A,B) + d(B,C) (triangle inequality).


A line  l  in an incidence geometry has a ruler if there exists
a function  r  from the set of points the points  l  contains
(or the point incident with  l , if one has choosen to speak that way)
to the set of real numbers  R, such that for points  A  and  B  on  l

                           d(A,B) = |r(A) - r(B)|
  
We call an incidence geometry with a distance function d a METRIC GEOMETRY
when it satisfies the RULER POSTULATE:  every line  l  has a ruler.



The PLANE SEPARATION AXIOM says that given a line  l  there exists two
subsets  H_1  and  H_2  of the plane  P  such that

 i)    every point of the plane P lies in exactly one of the sets l , H_1 or H_2.
 ii)   given two points A and B in H_1 we can connect them by a line segment
       in H_1, and ditto for H_2 (convexity of H_1 and H2);
 iii)  if a point  A  lies in H_1 and a point  B  lies in H_2 then the line
       segment  AB which connects them meets the line  l  .

We call a metric geometry which satisfies the Plane Separation Axiom a
PASCH GEOMETRY.

An angle measure (based on 180, say) consists of a function  m  from the set of all
angles to the set of real numbers such that

(i)   0 < m(angle ABC) < 180  for any angle  ABC ;
(ii)  for any number  t,  0< t < 180  and any ray  BC  on the edge of a 
      half-plane  H , there exists a unique ray  BA  with  m(angle ABC) = t ;
(iii) m(angle ADB) + m(angle DBC) = m(angle ABC)


We call a Pasch geometry together with an angle measure m a PROTRACTOR
GEOMETRY.


The SIDE-ANGLE-SIDE AXIOM states that given triangles ABC and
DEF such that 

  (i)   the length of side AB equals the length of side DE; 
  (ii)  the measure of angle B equals the measure of angle E; 
  (iii) the length of side BC equals the length of side EF;

we then have

  (ii)  the measure of angle C equals the measure of angle F; 
  (iii) the length of side AC equals the length of side DF;
  (ii)  the measure of angle A equals the measure of angle D; 

in other word, we have triangle ABC congruent to triangle DEF.


We call a protractor geometry which satisfies the Side-Angle-Side Axiom a
NEUTRAL GEOMETRY.


The EUCLIDEAN PARALLEL POSTULATE states that given a line l and a point P
not on l , there exists a unique line though P which lies parallel to l (in
other words, shares no point with l).


We call a neutral geometry that satisfies the Euclidean Parallel Postulate
a EUCLIDEAN GEOMETRY.