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\markright{MATH 110 -- Mathematics Through Time, Final Exam}

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\begin{document}

\noindent
Your Name:  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad}$

\vskip 10pt
\noindent
{\it Directions}
\vskip 10pt
\begin{itemize}
\item Do all work on the sheets provided 
\item If you use the back of a sheet,
please place a note telling me to look there.
\item There is an extra sheet of scratch paper at the end.  You may
detach that if you like, but please hand it in with your exam.
\end{itemize}
\vskip 30pt
Please do not write below on this page.
\vskip 30pt
\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
I  & \hfil/\  50\hfil \Strut \\
\hline
II  & \hfil/\  25\hfil \Strut \\
\hline
III or III' & \hfil/\  25\hfil \Strut \\
\hline
IV  & \hfil/\  25\hfil \Strut \\
\hline
V  & \hfil/\  25\hfil \Strut \\
\hline
VI (Essay)  & \hfil/\  50\hfil \Strut \\
\hline\hline
Total   & \hfil/ 200\hfil \Strut \\
\hline
\end{tabular}
$$\mathcal{HAPPY\ HOLIDAYS!}$$
\end{center}
\vfill\eject
\noindent
I.  Identifications.  For any 5 of the following historical figures, give the approximate historical 
period in which they lived, where they were active, and the main contributions they made that we discussed.
(If you submit answers for more than 5, only the best 5 will be counted.)
\vskip 10pt
\noindent
(A) (10)  Eratosthenes
\vskip 2.5truein
\noindent
(B) (10) Claudius Ptolemy
\vskip 2.5truein
\noindent
(C) (10) Muhmammad ibn Musa al-Khwarizmi
\vfill\eject
\noindent
(D) (10) Thabit ibn Qurra
\vskip 2.5truein
\noindent
(E) (10) Leonardo Pisano (``Fibonacci'')
\vskip 2.5truein
\noindent
(F) (10) Filippo Brunelleschi 
\vfill\eject
\noindent
II.
\begin{enumerate}
\item[(A)] (10) What are the 5 Common Notions (Axioms) and 5 Postulates that appear at the beginning
of Book I of Euclid's {\it Elements}?
\vskip 3.0truein
\item[(B)] (15) Proposition 32 in Book I deals with the sum of the angles
in a triangle.  What is the exact statement that Euclid proves?  Outline the construction and proof
that establishes this Proposition.  (You don't need to quote earlier propositions by number, but
do give a reason for each of your assertions.)
\end{enumerate}
\vfill\eject
\noindent
Choose either this problem (III) or III' on the next page.  (If you submit solutions for 
both, only the better of the two will be used.)
\begin{enumerate}
\item[(A)] (15)  What is true about the sum of the angles in a triangle under the alternate {\it hyperbolic}
form (H) of Postulate 5?  Outline the proof, assuming the needed
properties of Saccheri quadrilaterals.
\vskip 4.0truein
\noindent
\item[(B)] (10)  Was the result in part A the contradiction that Girolamo Saccheri, S. J. was seeking under
the assumption (H)?  Explain.   What did Janos Bolyai and Nikolai Ivanovich Lobachevsky eventually
show about the search for a contradiction starting from (H)?
\end{enumerate}
\vfill\eject
\noindent
III'.  (If you submit solutions for both this problem and III on the last page, only the better of the two will be used.)
\begin{enumerate}
\item[(A)] (15) The following problem appears in a text by the Indian mathematician
Mahavira (about 850 C.E.).  Solve it using our modern notation
and algebraic methods:  Of a collection of mangoes, the king took $1/6$, the queen
$1/5$ of the remainder, the chief princess $1/4, 1/3, 1/2$ of the successive
remainders, and the youngest child took the remaining $3$ mangoes.  Oh you who
are clever in miscellaneous problems on fractions, give the measure (number) of the
original collection of mangoes.  
\vskip 3.5truein
\item[(B)] (10)  What was probably the single largest contribution of ancient Indian mathematics
to the way we do mathematics today?  How and when did this contribution enter our mathematical
tradition?
\end{enumerate}
\vfill\eject
\noindent
IV.  
\begin{enumerate}
\item[A)] (15)  Draw the diagram for the Chinese ``{\it go gou} theorem" (with a general right triangle, 
sides $a,b$ and hypotenuse $c$) and show how it provides an algebraic/geometric proof of the 
general ``Pythagorean theorem.''
\vglue 4.0truein
\item[B)] (10) How is this proof different from the one given by Euclid in Proposition 47 of 
Book I of the {\it Elements}?
\end{enumerate}
\vfill\eject
\noindent
V. 
\begin{enumerate}
\item[A)]  (15) What are {\it perfect numbers}?  What are
{\it amicable pairs of numbers}?  Give an example of a perfect number and an amicable pair 
and explain how they satisfy the definitions.
\vglue 4.0truein
\item[B)] (10) How does the historical development of knowledge about these number-theoretical ideas
tie together the Greeks, the medieval Islamic mathematicians, and mathematics in the contemporary world?   
\end{enumerate}
\vfill\eject
\noindent
VI.  Essay.  (50)  Here are two contrasting statements about the ultimate legacy of
Greek mathematics:
\begin{itemize}
\item ``The death of Archimedes by the hands of a Roman soldier is symbolical of a
world-change of the first magnitude: the Greeks, with their love of abstract science, were
superseded ... by the practical Romans.  The Romans were a great race, but they were cursed
with the sterility which waits upon practicality.  They did not improve upon the knowledge of their forefathers,
and all their advances were confined to the minor technical details of engineering. They were not
dreamers enough to arrive at new points of view ... No Roman lost his life because he was absorbed
in the contemplation of a mathematical diagram.''  (Alfred North Whitehead -- 20th century philosopher)
\item
``There is no denying that the Greek approach to mathematics produced remarkable
results, but it also hampered the subsequent development of the subject. ... The Greek preoccupation
with geometry ... was a serious constraint.  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.'' (George G. Joseph, {\it The Crest of the
Peacock})
\end{itemize}
\noindent
Begin by summarizing and explaining your understanding of each statement. Then address
the following questions:  Have we seen examples in this class where practicality was {\it not sterile}, where 
concern with applications actually enriched the history of mathematics and made further progress possible?
Have we seen other examples where the ``abstract idealized constructs'' of pure mathematics led to major
advances?  Why do you suppose that Joseph doesn't mention Archimedes?  Does that omission weaken his point? 

\end{document}
