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\centerline{MONT 108N -- Mathematics Through Time}
\centerline{Problem Set 2 -- Egyptian Mathematics}
\centerline{{\bf due}: in class on September 27, 2010}
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\noindent
I.  Egyptian multplication and division
\item{A)} Multiply by successive doubling (that is, ``the Egyptian way''):  $434 \times 127$.
\item{B)} Multiply by successive doubling:  $252 \times 59$.
\item{C)} Divide by the Egyptian method:  $167 \div 18$.
\item{D)} What {\it advantage(s)} does the Egyptian method of multiplication
have over the method commonly used now?  What shortcomings?
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II.  Egyptian fractions -- the table of conversions of $2/n$ 
into unit fractions from the Rhind papyrus is given in 
modern notation on page 44 in {\it The Babylonian Theorem}.
Each row gives an expansion
$${2\over n} = {1\over a} + {1\over b} + {1\over c} + {1\over d}$$
for one of the odd integers $3 \le n \le 101$.  
(if there is no entry for $c$ and/or $d$, then those terms 
are not present).
\item{A)}  Verify that the rows for $n = 13$ and $n = 29$ are
correct.
\item{B)}  Use the table to find
a decomposition of ${4\over 27}$ into unit fractions.  
\item{C)}  Show that for any positive integer $n$, 
$${2\over n} = {1\over n} + {1\over 2n} + {1\over 3n} + {1 \over 6n}.$$
Yet this expansion appears only once in the table from the Rhind
papyrus.  For which $n$ did the compiler of the table fall back
on this form?  Why might they have preferred to avoid using this?
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III.  {\it The {\bf seqt} of a pyramid}.  The Egyptians measured the
steepness of a face of a pyramid by the ratio of the ``run''
(measured in hands) to the ``rise'' (measured in royal cubits).
There were $7$ hands in a royal cubit.  See page 32 in {\it 
The Babylonian Theorem}.  This ratio 
was called the {\it seqt} of the pyramid, measured in
units of {\it hands per royal cubit}.
\item{A)}  Problem 56 of the Rhind papyrus asks the solver
to find the {\it seqt} of a pyramid 250 royal cubits tall, 
with a square base 360 royal cubits on a side.  The answer
is given as $5 + {1\over 25}$ hands per royal cubit.  Is
this correct? 
\item{B)}  The great pyramid at Giza (built by the Pharaoh Khufu, finished
about 2560 B.C.E., and the only one surviving of the ``seven wonders of 
the ancient world'') has a base 440 royal cubits on a side and
a height of 280 royal cubits.  What is its {\it seqt}?
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IV. {\it An Egyptian approximation to $\pi$.}  We take $\pi$
to be the ratio between the area of a circle and the square
of the radius:  $\pi = {A\over r^2}$.
\item{A)}  In the Rhind papyrus, the area of a circle with
diameter $d$ is given as 
$$A_{circ} = \left({8\over 9} d\right)^2.$$
(This is not exactly correct, but yields decent approximate results.)  
What is the implied approximation to the value of $\pi$ here?
\item{B)}  Here's one way this may have been derived. Take
a square of side $9$ units, divide the edges into 
three equal pieces, and cut off four corner triangles 
with side length 3 to make an octagon.  The octagon 
is pretty close in area to the circle inscribed in
the original square.  Explain why the octagon has area
$63 \doteq 8^2$, and then explain how the approximation
$A_{circ} = \left({8\over 9} d\right)^2$ would result.  
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{\it Essay}
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Write about 1 page on your reactions to one of the following:
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\item{1)}  The talk by Tom Zetterstrom
\item{2)}  The photography exhibit of Tom Zetterstrom's work in the Cantor gallery
\item{3)}  The talk by John Hess on Darwin's trip to the Galapagos Islands
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In particular, did your experience of the talk or show affect how you view
the natural world and our place in it?  Be sure to state carefully
who the ``we'' is here -- humans generally, Americans,
Holy Cross students, some other group.    If so, how?  If not,
why not?  
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