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\centerline{MATH 392 -- Geometry Through History}
\centerline{Review Sheet for Midterm Exam}
\centerline{March 3, 2016}
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\noindent
{\it General Information}
\vskip 10pt
As announced in the course syllabus and the online schedule, the midterm exam 
will be given at the end of the first week after spring break.  If we decide to 
do the exam in class, it will be Friday, March 18.  I'm open to an evening time too, 
though, in case there is sentiment that a format allowing you more time to work would 
be preferable.  For instance, it would be possible to do the exam on Thursday
evening, March 17, say 7-9pm.  If we decided to do that, the normal Friday 
class would be cancelled.  We can discuss this in class on Friday, March 4.
(In any case, there will be no other written assignments due between now and
the exam.  So you should have ample time to prepare either way.)

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\noindent
{\it Format}
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The exam will cover the material on Euclidean and hyperbolic geometry that
we have studied since the start of the semester (i.e. Problem Sets 1 - 3 and
the first quiz).  There will be roughly equal coverage of the Euclidean and
hyperbolic cases.  

You should know the statements of Postulates I - V from 
Euclid, Pasch's Axiom, and definitions of terms we have used (especially
the \emph{revised} notion of parallelism and the angle of parallelism for the 
hyperbolic case, pencils of parallel lines, horocycles, horospheres.)  

Know the statement of the Bolyai-Lobachevsky theorem, and be prepared to use it 
to deduce facts about angles of parallelism in the hyperbolic case.

I will ask you to construct one or two short proofs you have not seen before (comparable 
to the proof from Quiz 1; also see the Review/Practice Problems below).  Other 
questions will ask for one or two of the more substantial assigned proofs 
listed below.  

I might also include a short answer question with several parts on aspects of the 
historical development, so you should know approximately when each of the following 
key figures lived, what their main contributions were, and how they figure into what 
we have done so far:
\begin{itemize}
\item Euclid of Alexandria
\item Proclus
\item Girolamo Saccheri, S.J.
\item Adrien-Marie Legendre
\item Janos Bolyai
\item Carl Friedrich Gauss
\item Nikolai Lobachevsky
\end{itemize} 

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\noindent
{\it Assigned Proofs}
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\begin{enumerate}
\item The proof of Proposition 29 in Book I of Euclid's {\it Elements}.  Also -- what
is especially notable about this Proposition?
\item The proof of Proposition 38 in Book I of the {\it Elements}, assuming Proposition 37.   
\item The proof of the Saccheri-Legendre Theorem (Theorem 2.2 in McCleary).  Also know
that this result holds in both Euclidean and hyperbolic geometry.  Why?
\item The proof that if $ABCD$ is a Saccheri quadrilateral on the base AB, then 
\begin{itemize} 
\item if HAA holds, then $AB < CD$ and the angle sum in any triangle is less
than 2 right angles, 
\item if HOA holds, then $AB > CD$ and the angle sum in any triangle is greater
than 2 right angles.  
\end{itemize}
(This is covered in Theorem 3.12 in McCleary.)  
\item The proof that the ratio of lengths of concentric arcs of two horocycles intercepted
by two lines in the parallel pencil is an exponential function of the distance between
the arcs.  (This is Theorem 4.9 in McCleary; we did the same proof in class as well.)
\end{enumerate}

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\noindent
{\it Review/Practice Problems}
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\begin{enumerate}
\item[(A)]  Assuming Postulate V, prove that the three internal angle bisectors of a triangle meet
in a single point.  (Where does your proof use Postulate V?)  Hint:  Take two bisectors that meet 
and join the intersection point to the third vertex.  Under what conditions does
that line bisect the angle at that third vertex?
\item[(B)]  Assuming Postulate V, prove that the three perpendicular bisectors of the sides of a 
triangle meet in a single point.  (Where did you use Postulate V?)  Hint:  You can use
the same strategy as in (A).
\item[(C)]  Look at the figure in problem 4.20 on page 70 in McCleary.  Show that in the 
hyperbolic plane, 
it is possible to construct three nonintersecting 
hyperbolic lines as in the top left diagram -- that is show
how to construct three lines $\ell_1, \ell_2$, and $\ell_3$ and points $A,A'$ on
$\ell_1$, $B,B'$ on $\ell_2$ and $C,C'$ on $\ell_3$ such that  
$\overrightarrow{AA'}$ is parallel to $\overrightarrow{BB'}$
in the direction of $A'$ and $B'$, and (at the same time) 
$\overrightarrow{B'B}$ is parallel to $\overrightarrow{CC'}$ in the direction of $B$ and $C'$, 
and (at the same time) $\overrightarrow{A'A}$ is parallel to $\overrightarrow{C'C}$
in the direction of $A$ and $C$.  Then generalize your argument to show that there
are similar ``$n$-gons'' of parallel lines for any $n \ge 3$.
\item[(D)]  Determine the derivative $\frac{d}{dx}(\Pi(x))$ and show that it is always
negative.
\item[(E)]  Let $x,y > 0$.  Express the angle of parallelism $\Pi(x+y)$ 
in terms of $\Pi(x)$ and $\Pi(y)$.
\end{enumerate}
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