\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,epsfig}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.25in}

\newcommand{\by}{\times}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\lcm}{lcm}
\newcommand{\Neg}{\phantom{-}}

\newcommand{\be}{\begin{enumerate}}
\newcommand{\ee}{\end{enumerate}}

\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}

\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}

\newcommand{\bs}{\begin{smallmatrix}}
\newcommand{\es}{\end{smallmatrix}}

\newcommand{\Answer}[1]{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|Answer}$}} 

\newcommand{\MyAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\ans}{\paragraph{Answer}}
\newcommand{\soln}{\paragraph{Solution}}
\newcommand{\sep}{{\medskip\hrule\vskip 10pt}}
\newcommand{\blank}{\underline{$\phantom{A blank}$}}

\newcommand{\vp}{$\vphantom{\big|}$}
\newcommand{\Vp}{$\vphantom{\Big|}$}

\newcommand{\ds}{\displaystyle}
\newcommand{\ts}{\textstyle}

%% Font declarations
\newcommand{\Number}[1]{\mathbf{#1}}
\newcommand{\Vector}[1]{\mathbf{#1}}
\newcommand{\Script}[1]{\mathcal{#1}}
 
%% Letter symbols
\newcommand{\C}{\Number{C}}
\newcommand{\N}{\Number{N}}
\newcommand{\Q}{\Number{Q}}
\newcommand{\R}{\Number{R}}
\newcommand{\Z}{\Number{Z}}
\newcommand{\e}{\Vector{e}}

\newcommand{\con}{\Script{C}} %% Conclusion of implication
\newcommand{\hyp}{\Script{H}} %% Hypothesis of implication  

\newcommand{\eps}{\varepsilon}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in}

\newcommand{\Blank}{\underline{\phantom{Answer}}} 
\newcommand{\Strut}{$\vphantom{\Bigg|}$}

\begin{document}
\centerline{MATH 110 -- Mathematics Through Time}
\centerline{Discussion 3 -- November 11, 2013}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
Today, we want to look at one important source of information about
early mathematics in India, the {\it Sulba Sutras}.  There are several
of these texts known with variations of different sorts.  The main idea
of all of them was the use of geometric relationships in the construction
of fire altars for the ancient Vedic religion.  The following information
is given in one of these texts for the construction of a particular 
great altar.

\begin{center}
\includegraphics[width=3truein,height=3truein]{SulbaCrop.png}
\end{center}

Our goal is to discover some of the numerical relationships 
occurring here and to interpret them in the light of one of the 
mathematical statements found in these texts.  
\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
\begin{enumerate}
\item[(A)]  Given that all of the triangles with only horizontal and slanted sides in the figure are
isosceles with symmetry across the line containing $X,Y$, 
show, using facts from Book I of the {\it Elements} that  
that $XY$ meets $AD$ and $BC$ at right angles at the points $P$ and $Y$.  Hint:  Use triangle congruences.
\item[(B)]  Now, given the information in the legend at the right of the figure, determine 
\begin{enumerate}
\item[1.]  The lengths $AP = PD$ (Hint:  what is true about the triangles $\Delta APX$ and $\Delta DPX$?)
\item[2.]  The lengths $AY = YD$ (note:  it's not entirely legible in the scanned image, but $XY = 36$)
\item[3.]  The lengths $CX = XB$
\item[4.]  The lengths $BR = RC$
\item[5.]  The lengths $AQ = QD$
\end{enumerate}
You should have noticed some intriguing patterns.  What is going on in this figure?  What 
is common to all of these pairs of triangles?  
\item[(C)]  In the text of all of the {\it Sulba Sutras} is the following statement:
{\it A rope stretched along the length of the diagonal of a rectangle makes an area
which the vertical and horizontal sides make together.}  How would we say this?  What
is this result?
\item[(D)]  (Short Essay/Thought Question) Recall the way H\o yrup's cut and paste interpretation of the steps 
on YBC 6967 (about 1800 BCE) is very close to one way of deriving the Pythagorean theorem.  That way
is essentially the same as the the figure for the {\it go-gou}
theorem from the Chinese {\it Zhou Bi Suan Jing}, which might date to 500 BCE or earlier.
All of the examples in (B) and the statement from (C) are from Indian texts thought to date from about 
800 BCE.    Proclus, writing about
Euclid's proof of the theorem in Book I of the {\it Elements} about 480 CE said, ``If we listen to those who
like to record antiquities, we shall find them attributing the theorem to Pythagoras and saying
that he sacrificed an ox on its discovery.  For my [i.e. Proclus's] part, though I marvel at those
who first noted the truth of this theorem, I admire more the author of the {\it Elements}, not 
only for the very lucid proof by which he made it fast [i.e. the proof of Proposition 47 in Book I], 
but also because he showed an even more
general statement in Book VI.''  Given all of this, why do you suppose 
we still call this result the {\it Pythagorean theorem}?
Do you admire more those who discovered the theorem or Euclid for finding such a systematic
and beautiful proof?
\end{enumerate}
\end{document}  

