Introduction to Neutral Geometry
( Sections 3.1, and 3.2 )
is the axiom system with the same undefined terms as those of Euclidean Geometry
but using the SMSG Postulates 1
15 as the only axioms.
One can think of Neutral Geometry as the
geometry which results from using as the axioms only Euclid‟s first
postulates (and those statements
he assumed as true without proof in his book “Elements” without listing them as postulates).
particular, Neutral Geometry ha
s no parallel postulate such as Euclid‟s 5
Postulate or such as the
SMSG Postulate 16 (Playfair‟s Postulate).
Since the SMSG Postulates 1
15 are axioms of Hyperbolic Geometry (as well as axioms of Euclidean
Geometry), the proofs of theorems in Neutral Geometry also prove that those statements are true in
Hyperbolic Geometry and they prove that those statements are true in Euclidean Geometry.
Similarly, these theorems proven using only Euclid‟s first four postulates are
true in both Euclidean
Geometry and Hyperbolic Geometry.
Among these first 15 SMSG Postulates are Postulate 4 (The Ruler Placement Postulate) and Postulate
12 (The Angle Construction Postulate).
In proofs, logical arguments based on Postulate 4 alone allow
one to conclude the existence and uniqueness (by SMSG Postulate 4) of points that are any specified
distance away from any specified point along any specified line in any specified direction along the line.
One can also conclude that, given two distinct points
A and B, there exists (by SMSG Postulate 4) a
unique distinct third point C such that the ratio (AC) / (CB) is equal to any specified positive real
In proofs in this class, assertions of the existence or uniqueness of points such as these may
have the phrase “by
Postulate 4” or the phrase “by the Ruler Placement Postulate” as their
Also, sections of proof based on SMSG Postulate 12 alone allow one to conclude the existence and
uniqueness (by SMSG Postulate 12) of angles of any specified measure with any specified ray as one side
of the angle with the other ray in a specified one of the two half-planes determined by the line containing
the specified ray. In proofs in this class, assertions of the existence or uniqueness of angles such as these
may have the phrase “by
Postulate 12” or the phrase “by the Angle Construction Postulate” as
their justifying reason.
These practices are illustrated in the proof of the following theorem:
Theorem 3.2.2: (i) Every line segment has exactly one midpoint;
(ii) Every angle has exactly one bisector.
Proof: (i) Let
be any line segment.
By SMSG Postulate 4, there exists a unique point
C on line
and between A and B such that
½ AB .
( ½ AB )
C is the unique midpoint of