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\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{MONT 107Q -- Thinking About Mathematics} \\
\textbf{Midterm Exam, March 31, 2017}
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\leftline{{\bf Your Name:}  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$ }
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\noindent
{\bf 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 blank sheet of paper at the end that you can 
use either as scratch paper or as extra space for your essay.  You may 
detach that if you like, but please put your name on it and hand it in with your exam
if you do detach it.
The numbers in parentheses next to the questions are there point
values (100 points total).

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\noindent\textbf{Please do not write in the space below}

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\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
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I    & \hfil/ 20\  \hfil \\
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II   & \hfil/ 20\  \hfil \\
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III  & \hfil/ 20\  \hfil \\
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IV   & \hfil/ 40\ \hfil\\
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Total   & \hfil/100\hfil \\
\hline
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\noindent
I.  On the Old Babylonian tablet YBC 7289 in Fig 1. above
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\includegraphics[width=.4\linewidth]{YBC7289c.jpg}
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\caption{Drawing of the YBC 7289 tablet by Asger Aaboe.  I do not have his permission to use this, but
he died in 2007, so I think the copyright issue is effectively moot(!)}
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\noindent
A) (10) The base-60 form of the number written directly over the diagonal of the 
square has just one base-60 digit to the left of the semicolon separating the whole number
part from the fractional part.  Give all the base-60 digits, separated by the semicolon and commas.
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\noindent
B) (10) Express the whole number from part A in base 10.  (It's OK to leave this as a sum of fractions
if you don't have a calculator.)
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\noindent 
II. 
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\noindent
A) (5) What does it mean to say that $\sqrt{2}$ and $1$ are \emph{incommensurable}?
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\noindent
B) (15) Give the proof the ancient Greeks found for the fact in part A.  What consequences
did this have for the way they thought of (pure) mathematics?
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\noindent
III.  Short answer.  Answer any four of the following.  If you answer
more than four, the best four will be used.
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\begin{enumerate}
\item[A)] (5)  What is the main feature that makes the presentation of the number theoretic
problems in Diophantus' {\it Arithmetica} different from earlier mathematics and a forerunner
of what we do now.
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\item[B)] (5)  What is the ``algebraic form'' of the fact proved in Proposition II.4
in Euclid's {\it Elements}? (The statement: If a straight line is cut at random then the square on the 
whole is equal to the squares on the pieces and twice the rectangle contained by the pieces.)
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\item[C)] (5)  What are the approximate dates of the YBC 6967 and YBC 7289 tablets?  About
when did Euclid live?
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\item[D)] (5)  What does the $M^o 20 \varsigma 1$ in Diophantus' Proposition I.4 mean?
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\item[E)] (5)  Explain what Al-Khwarizmi gave as a demonstration for his ``squares plus
roots equals numbers'' case of solving quadratic equations.  (Recall that he actually
did this in two different ways.  Either one is OK as an answer here; it is not necessary to
discuss both.)  
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\item[F)] (5)  How would the Old Babylonian scribes have approached a large calculation
such as multiplying $23,41;16$ by $17,28;30$?  (You don't need to do the calculation.)
Compare this with what we need to know to do multiplication in our base-10 positional number
system.
\end{enumerate}
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\noindent
IV.  Essay.  (40) You have the choice of responding to {\it either prompt 1 or 2}.  State which
one you have chosen at the start of your essay.
\begin{enumerate}
\item[1)]  ``The distinguishing feature of Babylonian mathematics is
its algebraic character.'' Of the historians we have mentioned, who
would agree with this claim, and who would disagree?  
Explain using the the interpretations your historians would give 
for the YBC 6967 problem of (what we would phrase as) solving the equation $x = 60/x + 7$.  
\item[2)]   George G. Joseph, the author of {\it The Crest of the Peacock}, offers
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 {\it hampered} [emphasis added] 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? 
Does this criticism seem to be apt for Diophantus' {\it Arithmetica}?
Then, which side of this debate do you come down on personally? Should all the mathematics 
we learn and do have practical usefulness or significance?
\end{enumerate}
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