% Math 370 hw2 Spring 2014
%
\documentclass{article}
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\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{verbatim}

%% create an environment for theorems
\newtheorem*{thm}{Theorem}

\newcommand{\coursehome}
{http://www.cs.umb.edu/~eb/370}

\title{History of Mathematics \\
Homework 2
}
\author{Ethan Bolker}

\begin{document}

\maketitle

This homework is due Thursday, February 13. I may add one question
based on the Lobachevsky reading.

\begin{enumerate}

\item Prove
\begin{thm}
The amount by which the sum of the angles of a spherical triangle
exceeds $\pi$ is proportional to the area of the triangle.
\end{thm}
\begin{proof}
Proof goes here, in a \verb!proof! environment. Note the ``QED''
symbol \qedsymbol{} (sometimes called a \emph{halmos})
that automatically appears at the end.

In the last page of his autobiography, Paul R. Halmos writes:

\begin{quotation}
    My most nearly immortal contributions are an abbreviation and a
    typographical symbol. I invented ``iff'', for ``if and only if''
    -�\ but I could never believe that I was really its first
    inventor. I am quite prepared to believe that it existed before
    me, but I don't know that it did, and my invention
    (re-invention?) of it is what spread it thorugh the mathematical
    world. The symbol is definitely not my invention �\- it appeared in
    popular magazines (not mathematical ones) before I adopted it,
    but, once again, I seem to have introduced it into mathematics. It
    is the symbol that sometimes looks like \qedsymbol{}, and is used to
    indicate an end, usually the  end of a proof. It is most
    frequently called the ``tombstone'', but at least one generous
    author referred to it as the ``halmos''.  
\end{quotation}

I found this at \url{http://jeff560.tripod.com/mathsym.html}.

\end{proof}

You can figure out your own proof, or find one somewhere and rewrite
it -- as long as you make it clear that you understand what you've
written. Don't just copy/paste. If you use work from elsewhere, tell
me where. 

You need not use historical methods. Modern tools are just fine.

\item Research the history of the theorem you just proved. Did the
  Greeks know it? Does it appear in Euclid's \emph{Elements}? 

\item Find out about the term \emph{pons asinorum}. You can start your
  google search at   Wikipedia, but don't just cut and paste from that
  page. I've read it  and don't need your reprise. Tell me something
  new and interesting.

\item Prove
\begin{thm}
If the sum of the angles of every right triangle is two right
angles then the sum of the angles of every triangle is two right angles.
\end{thm}

Hint (which you shouldn't need): you can cut an arbitrary triangle
into two right triangles.

\item Prove
\begin{thm}
If the sum of the angles of every triangle is two right angles then
Euclid's parallel postulate is true.
\end{thm}

If you can invent a proof of your own, do. But it's OK with me if you
look one up somewhere, understand it and present it. There's a link on
the course Euclid page that may help.

Proving that this hypothesis implies HRA in Sacchieri's quadrilateral
would suffice.

\end{enumerate}

\newpage

Here is the \LaTeX{} source for this document. You can cut it from the
pdf and use it to start your answers. I used the \verb!\jobname! macro
for the source file name, so you can call your file by any name you like.

\verbatiminput{\jobname}

\end{document}
