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{ \sc \large  APPENDICES: Something Hilbert Got Wrong and Euclid Got Right: The Method of  Superposition  and the Side-Angle-Side Axiom in Propositions 1 to 4 of Book I of the Elements }
 
February 10, 2012

\bigskip



 
   Eric B.  Rasmusen  

{\it Abstract}


\end{center}

\begin{small}

 This file just contains the appendices. 

 
 
\noindent
Rasmusen: Dan R. and Catherine M.
Dalton
Professor, Department of Business Economics and Public Policy, Kelley
School
of Business, Indiana University. BU 438, 1309 E. 10th Street,
Bloomington,
Indiana, 47405-1701. (812) 855-9219. Fax: 812-855-3354.
\url{mailto:erasmuse@indiana.edu}{ erasmuse@indiana.edu}, \url{http://www.rasmusen.org}.  

{\small
  \noindent This paper:
\url{http://www.rasmusen.org/papers/euclid-rasmusen-appendices.pdf}. }
 




\end{small}



\newpage
 
 
%--------------------------------------------------------


\refstepcounter{chaptercounter} \label{axioms} 


\noindent
{\sc Appendices: Systems of Axioms for Euclidean Geometry   }

This appendix will list various sets of axioms for Euclidean geometry--- Euclid's, Rasmusen's,   Hilbert's, Birkhoff's, Tarski's,  Veblen's and the SMSG's. 

\bigskip
 
\noindent
 {\sc Appendix I:  Euclid's Axioms }

 The definitions used in the postulates are boldfaced. 


\url{http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#posts}

 
\noindent
{\bf Definition 1.\\
    A point is that which has no part.

\noindent
Definition 2.\\
    A line is breadthless length.}

\noindent
Definition 3.\\
    The ends of a line are points.

\noindent
{\bf Definition 4.\\
    A straight line is a line which lies evenly with the points on itself.}

\noindent
Definition 5.\\
    A surface is that which has length and breadth only.

 

\noindent
Definition 7.\\
    A plane surface is a surface which lies evenly with the straight lines on itself.

\noindent
{\bf Definition 8.\\
    A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

\noindent
Definition 9.\\
    And when the lines containing the angle are straight, the angle is called rectilinear. 

\noindent
 Definition 10.\\
    When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.}

\noindent
Definition 11.\\
    An obtuse angle is an angle greater than a right angle.

\noindent
Definition 12.\\
    An acute angle is an angle less than a right angle.

\noindent
Definition 13.\\
    A boundary is that which is an extremity of anything.

\noindent
Definition 14.\\
    A figure is that which is contained by any boundary or boundaries.

\noindent
{
\bf Definition 15.\\
    A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.

\noindent
Definition 16.\\
    And the point is called the center of the circle.}

 

\noindent
Definition 19.\\
    Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

\noindent
Definition 20.\\
    Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

\noindent
Definition 21.\\
    Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

\noindent
  Definition 22.\\
    Of quadrilateral figures, a square is that which is both equilateral and right-angled;  an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

\noindent
  Definition 23\\
    Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.  

 
\noindent
Postulate 1. \\
To draw a straight line from any point to any point. 

\noindent
Postulate 2. \\
To produce a finite straight line continuously in a straight line.
 
\noindent
Postulate 3. \\
To describe a circle with any center and radius. 

\noindent
Postulate 4. \\
That all right angles equal one another. 

Postulate 5. \\
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. 

 

\noindent
Common notion 1.\\ 
Things which equal the same thing also equal one another.
 
\noindent
Common notion 2. \\
If equals are added to equals, then the wholes are equal. 

\noindent
Common notion 3. \\
If equals are subtracted from equals, then the remainders are equal. 

\noindent
Common notion 4. \\
Things which coincide with one another equal one another. 

\noindent
Common notion 5. \\
The whole is greater than the part. 

 Tarski and Given note that to decide the simplicity of an axiomatic system, you need to incorporate the definitions into it. That's right. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage


\noindent
 {\sc Rasmusen's Axioms },   

 
\noindent
Definition 1.\\
    A point is that which has no part.

\noindent
Definition 2.\\
    A curve is breadthless length.

\noindent
Definition 3.\\
    The ends of a curve are points.

\noindent
Definition 4.\\
    A segment is a curve which lies evenly with the points on itself.

\noindent
Definition 5.\\
    A surface is that which has length and breadth only.

 

\noindent
Definition 7.\\
    A plane surface is a surface which lies evenly with the segments on itself.

\noindent
Definition 8.\\
    A plane angle is the inclination to one another of two curves in a plane which meet one another and do not lie in a segment.

\noindent
Definition 9.\\
    And when the curve containing the angle are straight, the angle is called rectilinear.

\noindent
Definition 10.\\
    When a segment standing on a segment makes the adjacent angles equal to one another, each of the equal angles is right, and the segment standing on the other is called a perpendicular to that on which it stands.

\noindent
Definition 11.\\
    An obtuse angle is an angle greater than a right angle.

\noindent
Definition 12.\\
    An acute angle is an angle less than a right angle.

\noindent
Definition 13.\\
    A boundary is that which is an extremity of anything.

\noindent
Definition 14.\\
    A figure is that which is contained by any boundary or boundaries. [connected?]

\noindent
Definition 15.\\
    A circle is a plane figure contained by one line such that all the segments  falling upon it from one point among those lying within the figure equal one another.

\noindent
Definition 16.\\
    And the point is called the center of the circle.

 

\noindent
Definition 19.\\
    Rectilinear figures are those which are contained by segments, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four segments.

\noindent
Definition 20.\\
    Of trilateral figures, an equilateral triangle is that which has its three sides congruent, an isosceles triangle that which has two of its sides alone congruent, and a scalene triangle that which has its three sides noncongruent.

\noindent
Definition 21.\\
    Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

\noindent
Definition 22.\\
    Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

\noindent
Definition 23\\
    Parallel segments are segments which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. 

  \noindent
Postulate 1. \\
To draw a segment from any point to any point. 

\noindent
Postulate 2. \\
To produce a finite  segment continuously in a straight line.
 
\noindent
Postulate 3. \\
To describe a circle with any center and radius. 

 
 

\noindent
Common notion 1.\\ 
Things which equal the same thing also equal one another.
 
\noindent
Common notion 2. \\
If equals are added to equals, then the wholes are equal. 

\noindent
Common notion 3. \\
If equals are subtracted from equals, then the remainders are equal. 

\noindent
Common notion 4. \\
Things which coincide with one another equal one another. 

\noindent
Common notion 5. \\
The whole is greater than the part. 

Tarski and Given note that to decide the simplicity of an axiomatic system, you need to incorporate the definitions into it. That's right. 



Order of Propositions goes here too. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage


\noindent
 {\sc Hilbert's Axioms }, as taken from the 1950 English edition at \url{http://en.wikipedia.org/wiki/Hilbert%27s_axioms}. 


\noindent
{\bf I. Combination}\\

\noindent
1.	Two distinct points A and B always completely determine a straight line $a$. We write $AB = a$ or $BA = a$.

 Instead of ``determine,'' we may also employ other forms of expression; for example, we may say ``A lies upon $a$'', ``$A$ is a point of $a$'', ``$a$ goes through $A$ and through $B$'', ``$a$ joins $A$ to B'', etc. If A lies upon $a$ and at the same time upon another straight line $b$, we make use also of the expression: ``The straight lines $a$ and $b$ have the point $A$ in common,'' etc.

\noindent
2.	Any two distinct points of a straight line completely determine that line; that is, if $AB = a$ and $AC = a$, where $B  \neq C$, then also $BC = a$.

\noindent
3.	Three points A, B, C not situated in the same straight line always completely determine a plane $\alpha$. We write $ABC =  \alpha$.

 We employ also the expressions: ``A, B, C, lie in $\alpha$''; ``A, B, C are points of $\alpha$'', etc.

\noindent
4.	Any three points $A, B, C$ of a plane $\alpha$, which do not lie in the same straight line, completely determine that plane.

\noindent
5.	If two points A, B of a straight line a lie in a plane $\alpha$, then every point of a lies in $\alpha$. In this case we say: ``The straight line a lies in the plane $\alpha$,'' etc.

\noindent
6.	If two planes $\alpha$, $\beta$  have a point A in common, then they have at least a second point $B$ in common.

\noindent
7.	Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.

\bigskip
\noindent
 {\bf  II. Order}\\
\noindent
1.	If a point B is between points A and C, B is also between C and A, and there exists a line containing the points $A,B,C$.




 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{hilbert2-1.jpg} 
\begin{center}
{\sc Diagram \ref{hilbert2-1.jpg} \\
asdfsfsfd (axiom xxx)}
\includegraphics[width=\linewidth]{hilbert2-1.jpg}    
\end{center}
\end{wrapfigure}

\begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{hilbert2-2.jpg} 
\begin{center}
{\sc Diagram \ref{hilbert2-2.jpg} \\
asdfsfsfd (axiom xxx)}
\includegraphics[width=\linewidth]{hilbert2-2.jpg}    
\end{center}
\end{wrapfigure}




\noindent
2.	If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D.

\noindent
3.	Of any three points situated on a straight line, there is always one and only one which lies between the other two.

\noindent
4.	Pasch's Axiom:\\
  Let $A, B, C$ be three points not lying in the same straight line and let $a$ be a straight line lying in the plane ABC and not passing through any of the points $A, B, C$. Then, if the straight line $a$ passes through a point of the segment $AB$, it will also pass through either a point of the segment $BC$ or a point of the segment $AC$.


 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{hilbert2-5.jpg} 
\begin{center}
{\sc Diagram \ref{hilbert2-5.jpg} \\
asdfsfsfd (axiom xxx)}
\includegraphics[width=\linewidth]{hilbert2-5.jpg}    
\end{center}
\end{wrapfigure}

DEFINITION. xxx

\bigskip
\noindent
{\bf  III. Parallels}\\
\noindent
1.	In a plane $\alpha$ there can be drawn through any point A, lying outside of a straight line $a$, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A.
\bigskip

\noindent
{\bf   IV. Congruence}\\
1.	If $A, B$ are two points on a straight line $a$, and if $A'$ is a point upon the same or another straight line $a'$ , then, upon a given side of $A'$ on the straight line $a'$ , we can always find one and only one point $B$' so that the segment AB (or BA) is congruent to the segment $A'B'$ . We indicate this relation by writing $AB  \cong  A' B'$. Every segment is congruent to itself; that is, we always have $AB \cong AB$.
We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in one and only one way.

\noindent
2.	If a segment AB is congruent to the segment $A'B'$ and also to the segment $A''B''$, then the segment$ A'B'$ is congruent to the segment $A''B''$; that is, if $AB  \cong  A'B'$ and $AB  \cong  A''B''$, then $A'B'  \cong  A''B''$.

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\refstepcounter{figurecounter} \label{hilbert4-3.jpg} 
\begin{center}
{\sc Diagram \ref{hilbert4-3.jpg} \\
asdfsfsfd (axiom xxx)}
\includegraphics[width=\linewidth]{hilbert4-3.jpg}    
\end{center}
\end{wrapfigure}

\noindent
3.	Let AB and BC be two segments of a straight line a which have no points in common aside from the point $B$, and, furthermore, let $A'B'$ and $B'C'$ be two segments of the same or of another straight line $a'$ having, likewise, no point other than $B'$ in common. Then, if $AB  \cong  A'B'$ and $BC \cong  B'C'$, we have $AC  \cong  A'C'$.

DEFINITIONS: xxx

\noindent
4.	Let an angle $(h, k)$ be given in the plane $\alpha$ and let a straight line $a'$ be given in a plane $\alpha'$. Suppose also that, in the plane $\alpha'$, a definite side of the straight line $a'$ be assigned. Denote by $h'$ a half-ray of the straight line $a'$ emanating from a point $ O$' of this line. Then in the plane $\alpha'$ there is one and only one half-ray $k'$ such that the angle $(h, k)$, or $(k, h)$, is congruent to the angle $(h', k')$ and at the same time all interior points of the angle $(h', k')$ lie upon the given side of $a'$. We express this relation by means of the notation $ \measuredangle (h, k)  \cong   (h', k')$

Every angle is congruent to itself; that is, $ \measuredangle (h, k)  \cong  (h, k)$
or
$\measuredangle (h, k)  \cong  (k, h)$

\noindent
5.	If the angle $(h, k)$ is congruent to the angle $(h', k')$ and to the angle $(h'', k'')$, then the angle $(h', k')$ is congruent to the angle $(h'', k'')$; that is to say, if $\measuredangle (h, k)  \cong  (h', k')$ and $\measuredangle (h, k)  \cong  (h'', k'')$, then $\measuredangle (h', k')  \cong  (h'', k'')$.

\noindent
6.	If, in the two triangles $ABC$ and $A'B'C'$ the congruences $AB  \cong  A'B', AC  \cong  A'C', \measuredangle BAC  \cong  \measuredangle B'A'C'$ hold, then the congruences $\measuredangle ABC  \cong  \measuredangle A'B'C'$ and $\measuredangle ACB  \cong  \measuredangle A'C'B'$ also hold.

\bigskip



\noindent
{\bf   V. Continuity}\\
1.	Axiom of Archimedes.\\
 Let $A_1$ be any point upon a straight line between the arbitrarily chosen points A and B. Take the points $A_2, A_3, A_4, . . .$ so that $A_1$ lies between A and $A_2, A_2 $ between $A_1$ and $A_3, A_3$ between $A_2$ and $A_4$ etc. Moreover, let the segments $AA_1, A_1A_2, A_2A_3, A_3A_4, . . .$ be equal to one another. Then, among this series of points, there always exists a certain point An such that B lies between A and $A_n$.

\noindent
{ 2.	Line completeness}\\
 To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.

 
 	Hilbert (1899) included  II-4, but  
 R. L. Moore proved that this axiom is redundant, in 1902.

 The Parallel and Continuity axioms are sometimes omitted, as not necessary for a ``Hilbert plane'' and neutral geometry. Nor are the Continuity axioms necessary for Euclidean geometry, if certain propositions such as Proposition 1 are sacrificed and Euclid's proofs for certain other propositions are replaced.   (The present paper shows that III, 6 is also unnecessary, for Euclidean geometry). 


\newpage

\noindent
{\sc   Birkhoff's Axioms} \url{ http://en.wikipedia.org/wiki/Birkhoff%27s_axioms}

\noindent
Postulate I: 
 Postulate of Line Measure.\\
 A set of points $ \{A, B, ...\}$ on any line can be put into a 1:1 correspondence with the real numbers $\{a, b, ...\}$ so that  $|b - a| = d(A, B)$ for all points A and B.

\noindent
Postulate II: 
 Point-Line Postulate.\\
 There is one and only one line, $l$, that contains any two given distinct points P and $Q$.

\noindent
Postulate III: 
 Postulate of Angle Measure.\\
 A set of rays $\{  \mathpzc{l, m, n}, ...\}$ through any point O can be put into 1:1 correspondence with the real numbers $a (mod 2 \pi)$ so that if A and B are points (not equal to O) of $ \mathpzc{ l}$ and $\mathpzc{ m}$, respectively, the difference $a_\mathpzc{ m} - a_\mathpzc{ l} (mod 2\pi)$ of the numbers associated with the lines $\mathpzc{ l}$ and $\mathpzc{ m}$ is  $\measuredangle AOB$. Furthermore, if the point B on $\mathpzc{ m}$ varies continuously in a line $ r$ not containing the vertex O, the number $a_\mathpzc{ m}$ varies continuously also.

\noindent
{\bf Postulate IV: 
 Postulate of Similarity}\\
 Given two triangles $ABC$ and $A'B'C'$  and some constant $k > 0 , d(A', B' ) = kd(A, B), d(A', C' ) = kd(A, C)$ and  $ \measuredangle B'A'C'  = \pm  \measuredangle BAC$, then $d(B', C' ) = kd(B, C),  \measuredangle C'B'A'  =\pm \measuredangle CBA,$ and  $ \measuredangle A'C'B'  = \pm \measuredangle ACB$.


 Birkhoff's Postulate I says that the Euclidean metric is to be used, I think. Thus, Birkhoff can prove the Pythagorean Theorem using his 4 postulates.  And he can prove the parallel postulate.(See his 1932 paper.) 

I am puzzled that he thinks he needs his Postulate IV, SAS. He doesn't prove independence of his axioms. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\noindent
{\sc   Tarski's Axioms} (Tarski \& Given [1999])

$\rightarrow$,  $\leftrightarrow$, $\neg$, $\wedge$,  $\vee$.  $\bigwedge$,  $\bigvee$. $\exists$ $\forall$



 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski1.jpg} 
\begin{center}
{\sc Diagram \ref{tarski1.jpg} \\
Ax. 4. Axiom of Segment Construction}
\includegraphics[width=\linewidth]{tarski1.jpg}    
\end{center}
\end{wrapfigure}

\noindent
{\bf Ax. 1. Reflexivity Axiom for Equidistance}\\
$ab \equiv ba.$



 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski2.jpg} 
\begin{center}
{\sc Diagram \ref{tarski2.jpg} \\
 Ax. 5. Five-Segment Axiom}
\includegraphics[width=\linewidth]{tarski2.jpg}    
\end{center}
\end{wrapfigure}

\noindent
Ax. 2. Transitivity Axiom for Equidistance\\
$ab \equiv  pq \wedge ab \equiv  rs \rightarrow pq  \equiv  rs.$





 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski3.jpg} 
\begin{center}
{\sc Diagram \ref{tarski3.jpg} \\
Identity Axiom for Equidistance (ax 3)}
\includegraphics[width=\linewidth]{tarski3.jpg}    
\end{center}
\end{wrapfigure}
 
\noindent
Ax. 3. Identity Axiom for Equidistance\\
$ab \equiv cc \rightarrow a  = b.$
 



\noindent
Ax. 4. Axiom of Segment Construction\\
$ \exists x (B(qax) \wedge ax \equiv be).$




\noindent
Ax. 5. Five-Segment Axiom\\
$[a \neq b \wedge B(abc) \wedge B(a'b'c') \wedge ab \equiv a'b' \wedge bc \equiv b'c'\\
\wedge ad \equiv a'd' \wedge bd  \equiv b'd']  \rightarrow cd \equiv  c'd'. $


 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski6.jpg} 
\begin{center}
{\sc Diagram \ref{tarski6.jpg} \\
Identity Axiom for Betweenness (ax 6)}
\includegraphics[width=\linewidth]{tarski6.jpg}    
\end{center}
\end{wrapfigure}

\noindent
Ax. 6. Identity Axiom for Betweenness\\
$B(aba)  \rightarrow  a = b.$

\noindent
Ax. 7. First (or Inner) form of the Pasch Axiom\\
$B(apc) \wedge B(bqc)  \rightarrow  \exists x [B(pxb) \wedge B(qxa)].$

 

\noindent
Ax. 8(1. Lower 1-Dimensional Axiom\\
$\exists B \exists C (a \neq b).$

  \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski9.jpg} 
\begin{center}
{\sc Diagram \ref{tarski9.jpg} \\
 Upper O-Dimensional Axiom (ax9g)}
\includegraphics[width=\linewidth]{tarski9.jpg}    
\end{center}
\end{wrapfigure}

\noindent
Ax. $9^{(0)}$. Upper 0-Dimensional Axiom\\
$a =b.$



\noindent
Ax. 10. First Form of Euclid's Axiom\\
$B(adt) \wedge B(bdc) \wedge a \neq d  \rightarrow  \exists x \exists 3y [B(abx) \wedge B(acy) \wedge B(xty)].$

\noindent
Ax. 11. Axiom of Continuity\\
$\exists B \forall x \forall y [x \in X \wedge y \in  Y \rightarrow B(axy)]\\
 \rightarrow \exists b  \forall x \forall y [x \in  X \wedge y \in  Y  \rightarrow 
B(sby)].$

\begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski12.jpg} 
\begin{center}
{\sc Diagram \ref{tarski12.jpg} \\
Axiom of Continuity}
\includegraphics[width=\linewidth]{tarski12.jpg}    
\end{center}
\end{wrapfigure}




\noindent
Ax. 12. Reflexivity Axiom for Betweenness\\
$B(abb).$

\begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski13.jpg} 
\begin{center}
{\sc Diagram \ref{tarski13.jpg} \\
Transitivity Axiom for Betweenness}
\includegraphics[width=\linewidth]{tarski13.jpg}    
\end{center}
\end{wrapfigure}


\noindent
Ax. 13. $a =b  \rightarrow  B(aba).$


 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski14.jpg} 
\begin{center}
{\sc Diagram \ref{tarski14.jpg} \\
Symmetry Axiom for Betweenness (ax 14)}
\includegraphics[width=\linewidth]{tarski14.jpg}    
\end{center}
\end{wrapfigure}




\noindent
Ax. 14. Symmetry Axiom for Betweenness\\
$B(abc)  \rightarrow  B(cba).$

 
 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski15.jpg} 
\begin{center}
{\sc Diagram \ref{tarski15.jpg} \\
 xxx}
\includegraphics[width=\linewidth]{tarski15.jpg}    
\end{center}
\end{wrapfigure}



\noindent
Ax. 15. Inner Transitivity Axiom for Betweenness\\
$B(abd)  \wedge  B(bcd)  \rightarrow  B(abc)$.

\noindent
Ax. 16. Outer Transitivity Axiom for Betweenness
$B(abc)  \wedge  B(bcd)  \wedge  b \neq  c \rightarrow  B(abd).$

 


\noindent
Ax. 17. Inner Connectivity Axiom for Betweenness
$B(abd)  \wedge  B(acd)  \rightarrow  [B(abc) \vee B(acb)].$


 \begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski18.jpg} 
\begin{center}
{\sc Diagram \ref{tarski18.jpg} \\
Existence Axiom for Triangle Construction }
\includegraphics[width=\linewidth]{tarski18.jpg}    
\end{center}
\end{wrapfigure}


 
\noindent
Ax. 18. Outer Connectivity Axiom for Betweenness\\
$B(abc)  \wedge  B(abd)  \wedge  a \neq  b  \rightarrow  [B(acd) \vee B(adc)].$

\begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski19.jpg} 
\begin{center}
{\sc Diagram \ref{tarski19.jpg} \\
Tarski: sjkfjslj (ax 19)}
\includegraphics[width=\linewidth]{tarski19.jpg}    
\end{center}
\end{wrapfigure}


\noindent
Ax. 19. $a =b  \rightarrow ac \equiv bc.$

\begin{wrapfigure}{L}{2.6in}
\refstepcounter{figurecounter} \label{tarski20.jpg} 
\begin{center}
{\sc Diagram \ref{tarski20.jpg} \\
Uniqueness Axiom for Triangle Construction  (ax  20)}
\includegraphics[width=\linewidth]{tarski20.jpg}    
\end{center}
\end{wrapfigure}


\noindent
Ax. 20. Uniqueness Axiom for Triangle Construction\\
$[ac \equiv ac'  \wedge  bc \equiv bc'  \wedge  B(adb) \wedge  B(ad'b)  \wedge  B(cdx)  \wedge  B(c'd'x)\\
 \wedge  d\neq  x  \wedge  d' \neq x]  \rightarrow  c = c'. 
c$



\noindent
Ax. 21. Existence Axiom for Triangle Construction\\
$ab \equiv a'b'  \rightarrow  \exists c \exists x (ac \equiv  a'c'  \wedge  bc \equiv  b'c'  \wedge  B(cxp)\\
 \wedge  [B(abx) \vee  B(bxa)  \vee  B(xab)]).
c'$
 


\noindent
Ax. 22. Density Axiom for Betweenness\\
$x \neq z \rightarrow  \exists y [x \neq y  \wedge z  \neq  y  \wedge  B(xyz)].$
 
\noindent
Ax. 23. $[B(xyz)  \wedge  B(x'y'z')  \wedge  xy  \equiv x'y'  \wedge  yz \equiv y'z']\rightarrow xz \equiv  x'z'. 
$ 
 
  


\noindent
Ax. 24.$ B(xyz)  \wedge  B(x'y'z')  \wedge  xz  \equiv  xz'  \wedge  yz\equiv  y'z' -\rightarrow xy \equiv x'y'$

\margincomment{Maybe I need to add other n-dimesnatinoal versions of some of these aximos. ALso, some come in variants. }

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\newpage

\noindent
{\sc   Veblen's Axioms}

  The axioms are twelve in number; they presuppose only the validity of the
operations of logic and of counting (ordinal number). 

\noindent
Axiom I.\\
 There exist at least two distinct points.

\noindent
Axiom II.\\
 �� If points $A, B, C$ are in the order $ABC$, they are in the order
     CBA.

\noindent
Axiom HI.\\
    If points $A, B,   C$ are in the order $ ABC$, they are not in the
     order $B CA$.


\noindent
Axiom IV.\\
 If points $A, B, C$ are in the order $ABC$, then A is distinct
     from C.

\noindent
Axiom V.   \\
   If A and B are any two distinct points, there exists a point C such
     that A, B, C are in the order ABC.

\noindent
Def. 1.\\
 The line AB (A + B) consists of A and B and all points X in one
     of the possible orders ABX, AXB, XAB.                The points X in the order
     AXB constitute       the segment AB.      A and B are the end-points     of the
     segment.

\noindent
Axiom VI. \\
If points C and B (C + B) lie on the line AB, then A lies on
     the line CB.

\noindent
Axiom VII. \\
     If there exist three distinct  points, there exist three points A,
     B, C not in any of the orders ABC,         BCA, or CAB.

\noindent
Def.  2. \\
Three distinct points not lying on the same line are the vertices of a
     triangle ABC, whose sides ar� the segments AB, BC, CA, and whose
     boundary consists of its vertices and the points of its sides.

\noindent
Axiom VIII.    \\
    If three distinct points A , B, and C do not lie on the same
     line, and B and E are two points in the
     orders BCB        and CFA,      then a point F
     exists in the order AFB      and such that B,
     E, F lie on the same line.

\noindent
Def.   5.\\
 A point O is in the interior of a triangle if it lies on a segment, the end-points
     of which are points of different sides of the
     triangle.   The set of such points O is the
     interior of the triangle.

\noindent
Def. 6.\\
 If A, B, C form a triangle, the plane ABC             consists of all points
     collinear with any two points of the sides of the triangle.

\noindent
Axiom IX. \\
     If there exist three points not lying in the same line, there exists a
     plane ABC such that there is a point B not lying in the plane ABC.

\noindent
Def. 7.\\
 If A, B, C, and B are four points not lying in the same plane, they
     form a tetrahedron    ABCB      whose faces are the interiors of the triangles
      ABC, BCB,        CBA , BAB (if the triangles exist)* whose vertices are
     the four points, A, B, C, and D, and whose edges are the segments AB,
      BC, CB, BA, AC, BB.              The points of faces, edges, and vertices con-
     stitute the surface of the tetrahedron.

\noindent
Def. 8.\\
 If A, B, C, B are the vertices of a tetrahedron, the space AB CD
     consists of all points collinear with any two points of the faces of the tetra-
     hedron.



\noindent
Axiom X.    \\
  If there exist four points neither lying in the same line nor lying
     in the same plane, there exists a space AB CD such that there is no point
     E not collinear with two points of the space, AB CD.

\noindent
Axiom XI.    \\
   If there exists an infinitude of points, there exists a certain pair
     of points A C such that if [cr] * is any infinite set of segments of the line
     A C, having the property that each point which is A, C or a point of
     the segment A C is a point of a segment cr, then there is a finite subset
     cr1, a2, �� �, an with the same property.

\noindent
Axiom XII.      \\
  If a is any line of any plane a there is some point C o� a through
     which there is not more than one line of the plane a which does not inter-
     sect a.


We can do Crude Geometry without either the Parallel Postulate or the Triangle Postulate. This will not allow propositions about angles. But we can still do interesting things. I think the effect of getting rid of the Triangle Postulate is to allow local changes in curvature and metric, as in my example earlier.



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\newpage

\noindent
{\bf School Mathematics Study Group, Geometry. New Haven: Yale University Press, 1961.}

\noindent
Postulate 1. (Line Uniqueness) Given any two distinct points there is exactly one line that contains them.
     
\noindent
Postulate 2. (Distance Postulate) To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points.
     
\noindent
Postulate 3. (Ruler Postulate) The points of a line can be placed in a correspondence with the real numbers such that:

\noindent
        To every point of the line there corresponds exactly one real number.
        To every real number there corresponds exactly one point of the line.
        The distance between two distinct points is the absolute value of the difference of the corresponding real numbers.

 
\noindent
Postulate 4. (Ruler Placement Postulate) Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive.
     
\noindent
Postulate 5. (Existence of Points)

        Every plane contains at least three non-collinear points.
        Space contains at least four non-coplanar points.

 
\noindent
Postulate 6. (Points on a Line Lie in a Plane) If two points lie in a plane, then the line containing these points lies in the same plane.
  
\noindent   
Postulate 7. (Plane Uniqueness) Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.
    
\noindent 
Postulate 8. (Plane Intersection) If two planes intersect, then that intersection is a line.
     
\noindent
Postulate 9. (Plane Separation Postulate) Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that:

\noindent
        each of the sets is convex;
        if P is in one set and Q is in the other, then segment PQ intersects the line.

 
\noindent
Postulate 10. (Space Separation Postulate) The points of space that do not lie in a given plane form two sets such that:

\noindent
        Each of the sets is convex.
        If P is in one set and Q is in the other, then segment PQ intersects the plane.

 
\noindent
Postulate 11. (Angle Measurement Postulate) To every angle there corresponds a real number between 0� and 180�.
     
\noindent
Postulate 12. (Angle Construction Postulate) Let be a ray on the edge of the half-plane H. For every r between 0 and 180, there is exactly one with P in H such that $m�PAB = r.$
     
\noindent
Postulate 13. (Angle Addition Postulate) \\
If D is a point in the interior of �BAC, then $m�BAC = m�BAD + m�DAC.$
     
\noindent
Postulate 14. (Supplement Postulate) \\
If two angles form a linear pair, then they are supplementary.
     
\noindent
Postulate 15. (SAS Postulate) \\
Given a one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.
     
\noindent
Postulate 16. (Parallel Postulate)\\
 Through a given external point there is at most one line parallel to a given line.
     
\noindent
Postulate 17. (Area of Polygonal Region)\\
 To every polygonal region there corresponds a unique positive real number called the area.
     
\noindent
Postulate 18. (Area of Congruent Triangles)\\
 If two triangles are congruent, then the triangular regions have the same area.
     
\noindent
Postulate 19. (Summation of Areas of Regions) \\
Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2.
     
\noindent
Postulate 20. (Area of a Rectangle)\\
 The area of a rectangle is the product of the length of its base and the length of its altitude.
     
\noindent
Postulate 21. (Volume of Rectangular Parallelpiped)\\
 The volume of a rectangular parallelpiped is equal to the product of the length of its altitude and the area of its base.
     
\noindent\\
Postulate 22. (Cavalieri's Principle) Given two solids and a plane. If for every plane that intersects the solids and is parallel to the given plane the two intersections determine regions that have the same area, then the two solids have the same volume.

 




 
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