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\centerline{MATH 392 -- Geometry Through History}
\centerline{Syllabus -- Spring 2016}
\vskip 10pt
\leftline{Professor: John Little}
\leftline{Office: Swords Hall 331}
\leftline{Office Phone: (508) 793-2274}
\leftline{Office Hours:   M 2-3pm, T 10am-12noon, W 2-4pm, R 2-4pm, F 11am-12noon}
\leftline{$\qquad\qquad\qquad$ and by appointment}
\leftline{email: {\tt jlittle@holycross.edu}}
\leftline{Course Homepage:  {\tt http://mathcs.holycross.edu/\~{}little/GeometryThroughHistory/GTH.html}}
\vskip 10pt
\begin{verse}
Euclid alone has looked on Beauty bare.\\
Let all who prate of Beauty hold their peace,\\
And lay them prone upon the earth and cease\\
To ponder on themselves, the while they stare\\
At nothing, intricately drawn nowhere\\
In shapes of shifting lineage; let geese\\
Gabble and hiss, but heroes seek release\\
From dusty bondage into luminous air.\\
$ $      \\
O blinding hour, O holy, terrible day,\\
When first the shaft into his vision shone\\
Of light anatomized! Euclid alone\\
Has looked on Beauty bare. Fortunate they\\
Who, though once only and then but far away,\\
Have heard her massive sandal set on stone.\\
$ $      \\
$\qquad\qquad\qquad$\emph{Edna St. Vincent Millay, 1892-1950}\\ 
\end{verse}
\vskip 10pt
\noindent
{\it Course Description}
\vskip 10pt
Geometry has been a central part of mathematics since the time of the ancient
Greeks.  The understanding of plane and solid geometry achieved and codified in
the {\it Elements} of Euclid (ca. 300 BCE) has remained, ever since, in one form or 
another, both a model for the presentation of mathematics and an important part of 
mathematical education.   In modern research mathematics, too, subjects
known as {\it differential geometry} and {\it algebraic geometry} are  
active and dynamic fields of study.  The content of those modern subjects 
can seem very different because the connections with the past are not always apparent. 
When viewed through a historical lens, though, we see a fascinating story of how 
questions raised by close study of Euclid motivated new tools and new approaches that 
led directly to these later incarnations of geometry. 

In this course, we will follow some of that historical development to 
identify and trace those fascinating connections through time.   
\begin{itemize}
\item We will begin ``at the beginning,'' with Euclid's geometry (especially Book 
I of the {\it Elements}), which will be familiar to most of you through your high 
school mathematical course work.  We will focus there on the role of Euclid's 
{\it Postulate V} in the theory of {\it parallels}.  
\item We will note that it is possible to ``do geometry'' 
on the surface of the (two-dimensional) sphere and that the resulting spherical geometry
is actually quite different from plane geometry.  (This makes it somewhat hard to fathom part
of the subsequent history because Greek mathematicians after the time of 
Euclid were very familiar with properties of spherical triangles and other aspects of 
that geometry!)  
\item We will see that after many attempts to ``vindicate'' Euclid by clarifying the 
status of Postulate V, in the early 19th century CE, through work of Janos Bolyai, 
Nikolai Lobachevsky (and Carl Friedrich Gauss), it became clear that there were
alternate, {\it non-Euclidean} geometries--perfectly consistent mathematical 
theories on an equal footing with Euclidean geometry, but in which Postulate V 
does not hold.
\item Starting with the work of Ren\'e Descartes in the 16th century CE, and the 
subsequent development of calculus by Isaac Newton and Gottfried Leibniz, the
use of real numbers to measure lengths and describe locations of points through 
coordinate geometry provided a collection of different ways
to define geometric objects and study their properties. 
\item Work by Gauss on the intrinsic geometry of surfaces and Bernhard Riemann's 
{\it Habilitationsvortrag} led to a very general way of understanding geometry 
that showed how non-Euclidean geometries could be realized and opened the way 
to applications such as Albert Einstein's general theory of relativity.  
\end{itemize}  
\vskip 10pt
\noindent
{\it Course Objectives}
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\noindent
The major objectives of the course will be:
\vskip 10pt
\begin{enumerate}
\item[1)] To examine the history and development of geometry through time, 
using some of the original sources to understand how and why those ideas were developed.
\item[2)] To make connections between different parts of your mathematical studies
and serve as a ``capstone'' to your major program
\item[3)] To give you the experience of working on a substantial mathematical {\it project}
leading to a written paper and oral presentation.  
\vskip 10pt
\end{enumerate}
\noindent
{\it Texts}
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\noindent
The main text book for the course is:
\begin{enumerate}
\item[] J. McCleary, {\it Geometry from a Differentiable Viewpoint}, 2nd edition, 
Cambridge University Press.
\end{enumerate}
A few other shorter readings will be distributed as .pdf files via the course homepage. 
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\noindent
{\it Course Schedule}
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A detailed day-by-day course schedule with due dates for assignments
will be maintained on the course homepage.  That listing is a tentative, evolving schedule,
so you will probably want to refer to it frequently.  Any important changes will also
be announced in class well in advance.   
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\noindent
{\it Assignments and Grading}
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\begin{enumerate}
\item[1)]  Midterm Exam (20 \% of course grade) -- tentative date: Friday, March 18.
\item[2)]  2 shorter quizzes (10 \% of course grade) -- tentative dates:  Friday, February 19 and
Friday, April 15
\item[3)]  Final Project -- paper (25 \% of course grade), due:  Monday, May 9,
and oral presentation (15\% of course grade) given in class, weeks of April 25 and May 2 
(schedule to be determined later) 
\item[4)]  Problem Sets (about 8 through the semester -- 20 \% of course grade)
\item[5)]  In-class group work (about 5 assignments -- 10 \% of course grade)   
\end{enumerate}
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I will be keeping your course average in numerical form throughout the semester, and 
only converting to a letter for the final course grade. The course grade will be 
assigned according to the following conversion table (also see Note below):
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\begin{itemize}
\item A -- 94 and above
\item A- -- 90 - 93
\item B+ -- 87 - 89
\item B -- 84 - 86
\item B- -- 80 - 83
\item C+ -- 77 - 79
\item C -- 74 - 76
\item C- -- 70 - 73
\item D+ -- 67 -- 69
\item D -- 60 - 66
\item F -- 59 and below.
\end{itemize}
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\noindent
Note: Depending on how the class as a whole is doing, some downward 
adjustments of the above letter grade boundaries may be made. No upward 
adjustments will be made, however. (This means, for instance, that an 85 
course average would never convert to a letter grade of B- or below, 
although it might be a B+ in some circumstances.) If you ever have a 
question about the grading policy or your standing in the course, don't 
hesitate to ask me.
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\noindent
{\it Advice On How To Succeed In This Class}
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\noindent
{\bf A good ``work ethic" is key.}  As you should be able to tell
from the course description above, you do not need to be a 
``math genius'' to do well in this course.  But you will need 
to put in a consistent effort and keep up with the reading and 
assignments.

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\noindent
{\bf Come to class.}  Unless you are deathly ill, have a genuine family 
emergency, are away at a game or meet of a college athletic team, etc. 
plan on showing up here at 2:00 pm every Monday, Wednesday, 
and Friday this semester.  Many of the class meetings will be 
structured around discussions or student presentations.  Your 
participation is expected and needed for the success of the course!

\vskip 10pt
\noindent
{\bf Take notes and use them.} This may seem obvious, but it is worth 
saying!  Used intelligently, your notes can be a valuable resource as 
you work on problem sets and prepare for exams.

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\noindent
{\bf Use the text and class notes actively.}  Reading about mathematics
is not like reading a novel.  You will probably need to read and think over things 
more than once.  You may want to work through examples or draw your own 
diagrams to understand some of the Euclidean proofs that we do.  

\vskip 10pt
\noindent
{\bf Set up a regular study schedule and work at a steady pace.} It's not easy 
to play catch-up in a mathematics course (even when the course is part of a 
first-year program with additional goals beyond the mathematics).  You should 
expect to budget at least 6 hours in a typical week for work outside of class. 
The best way to use your time is to do a few problems, some reading from the 
books, and reviewing of class notes every day.

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\noindent
{\bf Most importantly, if you are having difficulty learning something, 
get help as soon as possible.}  You can do this by asking questions during 
class (any time something isn't clear), or seeing me during office hours.

\vskip 10pt
\noindent
{\it Statement on Academic Integrity}
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All education is a cooperative enterprise between teachers and students.  This 
cooperation works well only when there is trust and mutual respect between everyone 
involved.  To be become an engaged and advanced learner, you must be able to think 
and work both independently and in concert with your peers.  
 
The College academic honesty policy states: 
``As an institution devoted to teaching, learning, and intellectual inquiry, Holy 
Cross expects all members of the College community to abide by the highest standards 
of academic integrity. Any violation of academic honesty undermines the student-teacher 
relationship, thereby wounding the whole community. The principal violations of academic 
honesty are plagiarism, cheating, and collusion.

Plagiarism is the act of taking the words, ideas, data, illustrative material, or 
statements of someone else, without full and proper acknowledgment, and presenting them 
as one's own.

Cheating is the use of improper means or subterfuge to gain credit or advantage. Forms 
of cheating include the use, attempted use, or improper possession of unauthorized aids 
in any examination or other academic exercise submitted for evaluation; the fabrication 
or falsification of data; misrepresentation of academic or extracurricular credentials; 
and deceitful performance on placement examinations. It is also cheating to submit the 
same work for credit in more than one course, except as authorized in advance by the 
course instructors.

Collusion is assisting or attempting to assist another student in an act of academic 
dishonesty.''

The full statement on Academic Honesty in the College Catalog  is available at
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\centerline{\tt http://www.holycross.edu/catalog/academic-honesty-policy.pdf}
\vskip 10pt
\noindent
While the temptation to engage in an act of academic dishonesty may arise because of
time pressure, other events in your life, and so forth, using dishonest means to enhance 
a single grade is not worth the loss of your personal integrity.  
If in doubt about what you plan to do or write violates academic honesty, 
PLEASE ASK!  If you do not know how to correctly cite reference materials, consult with me, 
the Writer’s Workshop, and/or visit the link below.

\noindent
Department of History: 
\vskip 10pt
\centerline{\tt http://www.holycross.edu/departments/history/website/academichonesty.htm}


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\noindent
{\it Specific Guidelines for this Course}
\vskip 10pt
In this course, the examinations will be closed-book. No 
sharing of information with other students in any form will be permitted during exams. 
On group discussion write-ups, close collaboration with the other members of your group 
is expected. On the individual problem sets, discussion of the questions with other 
students in the class and with me during office hours is allowed, 
{\it even encouraged}.  However, your final problem solutions should 
be prepared individually and the wording and organization of your final 
problem solutions should be entirely your own work.  Moreover, if you do 
take advantage of any of the above options for discussion of problems with 
others, you will be required to state that fact in a footnote accompanying 
the problem solution. Failure to follow this rule will be treated as a violation 
of the College's Academic Integrity policy.  For the papers, 
if you do consult a source other than the course texts, include a full 
reference in a bibliography section at the end of your paper, and identify 
any direct quotations.  Information about the acceptable formats for doing 
this will be distributed with the paper assignments.  
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