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\leftline{MONT 104Q -- Mathematical Journeys}
\leftline{Midterm Exam, November 2, 2015}
\leftline{Your Name:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$ }
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\noindent
{\it Directions}
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Do all work on the sheets provided (if you use the back of a sheet,
please place a note telling me to look there).  
There is an extra sheet of scratch paper at the end.  There are 100 possible points,
distributed as indicated in the questions.
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\noindent
I.  
\begin{enumerate}
\item[(A)]  (10) G. H. Hardy included two theorems and proofs from Euclid's {\it Elements} as
prime examples of ``real, serious, beautiful mathematics'' in his book 
{\it A Mathematician's Apology}.  Give the statements of those two theorems.  
{\it You do not need to supply the proofs for these}.
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\item[]  First theorem:
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\item[] Second theorem:
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\item[(B)]  (5)  Hardy says ``the mathematics which has permanent aesthetic value ... may
continue to cause intense emotional satisfaction to thousands of people after thousands of 
years.''   About how much time (to within 100 years) separates the time of Euclid from Hardy's
own time? 
\end{enumerate} 
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\noindent
II.  
\begin{enumerate}
\item[(A)] (10)  State the 5 Common Notions (Axioms) and 5 Postulates at the start of 
Book I of the {\it Elements} of Euclid.  
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\item[(B)]  (5)  What is the purpose of the statements from part A in Euclid's logical
scheme?  How are the Common Notions different from the Postulates?
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\end{enumerate}
\noindent
III.  (10) Proposition 1 in Book I of Euclid's {\it Elements} is a construction for 
an equilateral triangle with side equal to any given line segment AB.  Give the construction
and the proof that the construction is valid, with justifications for each of the steps in 
the proof based on your answers to question II.  
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\noindent
IV.  Proposition 47 in Book I of the {\it Elements} is a famous statement from geometry
illustrated by the figure above.  Use the labeling here in your answers to all parts.
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{EuclidI47.jpg}
\caption{Figure for Proposition 47, Book I}
\end{center}
\end{figure}
\begin{enumerate}
\item[(A)]  (5)  Give the statement {\it in Euclid's form} and the usual name of this result. 
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\item[(B)] (5)  How is the dotted line $AM$ in the figure constructed?
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\item[(C)] (5)  In the first part of the proof, Euclid shows that $\Delta GBF$ has the same
area as what other triangle in the figure?  Why does that follow?
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\item[(D)] (5)  The second part of the proof consists of showing that $\Delta FBC$ and 
$\Delta ABD$ are congruent.  How does that follow?  (Show that is true using one of the 
triangle congruence results proved before in Book I.)  
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\item[(E)] (5)  How does Euclid conclude that $ABFG$ and $BLMD$ have the same area?  And
how does he conclude the proof?
\end{enumerate}
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\noindent
IV.  Essay.  (35) George G. Joseph, the author of an interesting book about the history of mathematics called {\it The Crest of the Peacock}, offered this overall evaluation of the ultimate impact of Greek geometry:  ``There is no denying that the Greek approach to mathematics produced 
remarkable results, but it also hampered the subsequent development of the
subject.  ...  Great minds such as Pythagoras, Euclid, and Apollonius spent much
of their time creating what were essentially abstract idealized constructs; how they 
arrived at a conclusion was in some way more important than any practical significance.'' 
First, what does the last sentence mean?   What is Joseph getting at? 
Then, based on what he said in {\it A Mathematician’s Apology}, how would G.H. Hardy respond
to Joseph?  Finally, which side of this debate do you come down on personally? Should all the mathematics we learn and do have practical usefulness or significance?
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$\qquad$
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