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\textbf{College of the Holy Cross, Fall 2019} \\
\textbf{MATH 110-02 -- Algebra Through History} \\
\textbf{Midterm Exam, October 25, 2019}
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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|}
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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
\end{tabular}
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\noindent
I. 
\begin{figure}
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\includegraphics[width=.4\linewidth]{YBC7302.png}
\caption{The Old Babylonian tablet YBC 7302}
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\noindent
A) (15) Refer to Figure 1 above showing an Old Babylonian practice tablet
from a scribal school in the city of Susa.  It shows three separate numbers
in the cuneiform base-60 script.  What are those three numbers?
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\noindent
B) (5) From the context (the rough circle with the numbers), 
it is surmised that this shows a calculation of an approximate area of a circle. 
Interpret the number written over the circle as the circumference, and the number
inside the circle as the area.  {\it Question:} What value is being used for 
the constant $\pi$?  Hints:  (1)  The number inside is not a whole number; you want to 
think of it as the first digit after the $;$ in the fractional part.  Useful information:
the area of a circle of radius $r$ is $A = \pi r^2$ and the circumference is $C = 2\pi r$, 
so the area is $A = \frac{C^2}{4\pi}$ in terms of the circumference.    
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\noindent 
II. 
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\begin{enumerate}
\item[A)] (5) In Greek mathematics, what did it mean to say that that two magnitudes
are \emph{incommensurable}?  
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\item[B)] (15) Give the proof the ancient Greeks found for the fact that 
the diagonal of a square of side 1 is not commensurable with the side.
\end{enumerate}  
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\noindent
III.  Short answer.  Answer any four of the following.  If you answer
more than four, you can earn some Extra Credit points.
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\begin{enumerate}
\item[A)] (5)  One of Diophantos' propositions contains the following expression 
$\Delta^\Upsilon 4 M^o 16 \Lambda \s 16$ (containing one unknown).  Rewrite this in 
modern notation, calling the unknown $x$.  
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\item[B)] (5)  What are the approximate dates of the YBC 6967 and YBC 7289 tablets?  About
when did Euclid live?
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\item[C)] (5)  What is ``Fermat's Last Theorem'' and what is its connection with Diophantos?
When and by whom was this finally solved?
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\item[D)] (5) In what way is the solution of the problem given on the YBC 6967
tablet related to our modern quadratic formula?
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\item[E)] (5)  Proposition 4 in Book II of Euclid's \emph{Elements} says:  {\it If a 
straight line is cut at random then the square on the whole line is equal (in area) to the sum of
the squares on the pieces together with twice the rectangle contained by the two pieces.}
What modern algebraic equation is equivalent to this?  Call the lengths of the two pieces $x,y$.
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\item[F)] (5) What would be the hardest arithmetic operations to carry out in the Old
Babylonian base-60 number system?  How did they keep track of all the information they needed for these?
\end{enumerate}
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\noindent
IV. (40) Essay.  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)]  A certain older history of mathematics says, flatly, that  
``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 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 another book on the non-European roots of 
modern mathematics called {\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} 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?  
Would this criticism seem to be apt for 
Diophantos' {\it Arithmetica} as well?  Is it necessary for all the mathematics 
we learn and do to have practical usefulness or significance?  
\end{enumerate}
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