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\begin{document}

\title{History of mathematics}
\subtitle{Log of a course}
\author{David Pierce}
\date{July 30, 2010}

\uppertitleback{\center{
This work is licensed under the\\
 Creative Commons Attribution--Noncommercial--Share-Alike
License.\\
 To view a copy of this license, visit\\
  \url{http://creativecommons.org/licenses/by-nc-sa/3.0/}\\
\mbox{}\\
\cc \ccby David Pierce \ccnc \ccsa\\
\mbox{}\\
Mathematics Department\\
Middle East Technical University\\
Ankara 06531 Turkey\\
\url{http://metu.edu.tr/~dpierce/}\\
\url{dpierce@metu.edu.tr}
}
}

\maketitle%[-1]

\tableofcontents

\listoffigures

\addchap{Prolegomena}

\addsec{What is here}

This book is a record of a course in the history of
mathematics, held at METU during the 2009/10
academic year.  Officially the course 
was 
\begin{compactenum}[(1)]
  \item
Math 303, History of Mathematical Concepts I, in the fall
semester; 
\item
Math 304, History of Mathematical Concepts II, in the
spring.  
\end{compactenum}
There were about twenty students in each semester; but only
four students took both semesters.  The two semesters correspond to
the two numbered parts of this book.  
According to the catalogue, the course content is thus:
\begin{quote}
  [Math 303:] Mathematics in Egypt and Mesopotamia, Ionia and Pythagoreans,
  paradoxes of Zeno and the heroic age. Mathematical works of Plato,
  Aristotle, Euclid of Alexandria, Archimedes, Apollonius and
  Diophantus. Mathematics in China and India.  [Math 304:] 
Mathematics of the Renaissance, Islamic contributions. Solution of the
cubic equation and consequences. Invention of logarithms. Time of
Fermat and Descartes. Development of the limit concept. Newton and
Leibniz. The age of Euler. Contributions of Gauss and
Cauchy. Non-Euclidean geometries. The arithmetization of analysis. The
rise of abstract algebra. Aspects of the twentieth century.  
\end{quote}
Most parts of this description correspond to chapter titles in the
suggested textbook by Boyer~\cite{MR996888}.  But I did not use a modern
textbook.  My way of teaching the course was inspired
by my experience at St.\ John's College, with campuses in
Annapolis, Maryland, and Santa Fe, New Mexico, USA.  As a student at
St.\ John's, I 
learned mathematics by reading, presenting, and discussing the works
of Euclid, Apollonius, Descartes, Newton, and others.  In teaching
Math 303--304, I hoped my own students could learn in the same way.  So my
course had no textbook other than the works (in English translation)
of the mathematicians that we studied.  In class, students presented
the content of these works at the blackboard. 

My notes\label{Johnnies} in
Part~\ref{part:one} below
started out as emails to a discussion group, the `J-list', comprising
St John's alumni. 
The dates used as section heads in this part are the original dates of
composition of these emails; but I have done some editing.

In the spring semester, the
conversion of emails into \LaTeX\ (so that they could be incorporated
in a book such as this one) became too tedious; also I wanted to
use diagrams; so I started 
composing these notes directly in \LaTeX.  In
Part~\ref{part:two} of this book, section titles are simply dates of classes.

A big difference between courses at St John's College and courses at METU is
that the latter have written examinations.  Those exams that I wrote
for Math 303--304 are in Appendix~\ref{app:exams}.

Whether the course was a success might be judged from student
comments, which I invited on the final exams; these are in
Appendix~\ref{app:comments}. 

On the other hand, students are not necessarily the best judges of
their own progress.  It is also the case that one of the best and most
enthusiastic students, Mehmet D., did not write me any comments;
below I shall mention some of what he told me face to face.
Meanwhile, 
I judge the course to have been successful, at least insofar as it taught
students that they \emph{could} read some of the great works of
mathematics.  As can be seen from their comments, some of the students
wished I had just 
\emph{told} them what was in those books.  If the course had been simply a
mathematics course, I could have done that.  But the course was a \emph{history}
course, and the whole point of history is to understand what people in
the past have \emph{thought.}  In saying this (and I shall say more
about it below), I am following the Oxford 
philosopher R. G. Collingwood (1889--1943), some of whose remarks on
history are in Appendix~\ref{app:Col}.

My attempts to communicate to my department what I was doing with the
course are in Appendix~\ref{app:cor}, along with the responses of the
sole person who did respond.

Appendix~\ref{app:background} consists of some notes on ancient Greek
mathematics that I put on the webpage of Math 303 at the beginning of
the year. 

\addsec{Apology}

If I were to teach Math 303--304 again (which I should like to do),
then I should 
certainly make some changes.  But the practice of reading and
presenting original sources, especially older ones, ought to be
maintained, for reasons including the following. 

\minisec{Scientific history}

Studying history does not mean learning to express \emph{opinions}
about what people of the past thought; it is learning what they
thought.  In saying this, I have in mind the distinction between
opinion and knowledge expressed by the character of Socrates in
Plato's \emph{Republic} \cite[II, p.~92; 506C]{Shorey}: 
\begin{quote}
Have you not observed that opinions (\Gk{d'oxai}) divorced from
knowledge (\Gk{epist'hmh}) are ugly things?\footnote{\Gk{o>uk
    >'h|sjhsai t`as >'aneu >epist'hmhs d'oxas, <ws p~asai a>isqra'i?}} 
\end{quote}
A teacher can \emph{tell}
students what he believes Euclid thought, and the students can learn
to repeat these teachings; but the teachings are only opinions for the
students, if not for the teacher, unless the students test the
opinions against what Euclid actually wrote.   

A teacher's lectures on
math history may be useful for students' \emph{mathematics.}  In
\emph{A Comprehensive Introduction to Differential
  Geometry}~\cite[p.~vi]{MR82g:53003a}, Spivak writes, 
\begin{quote}
  Of course, I do not think that one should follow all the intricacies
  of the historical process, with its inevitable duplications and
  false leads.  What is intended, rather, is a presentation of the
  subject along the lines which its development \emph{might} have
  followed; as Bernard Morin said to me, there is no reason, in
  mathematics any more than in biology, why ontogeny must recapitulate
  phylogeny.  When modern terminology finally is introduced, it should
  be as an outgrowth of this (mythical) historical development.
\end{quote}
Spivak here is getting ready to teach mathematics, not history.  It is
useful for him and his readers to look at the 
history of the mathematics; but then that history will be adapted to
the needs of the mathematics.  In this case, as Spivak suggests,
history becomes a myth, a kind of story.  It may be an enjoyable or
useful story.  The story may be based on historical \emph{knowledge}
on the part of the storyteller-mathematician.  But then the story is not
designed to share all of that knowledge.  For the listener or reader
then, the  story---the myth---can only be a kind of \emph{opinion,} in
the sense of Plato.  It is no longer history.  

In \emph{The Principles of History}~\cite[pp.~12 f.]{Collingwood-PH},
Collingwood derides what he calls 
`scissors-and-paste' history:
\begin{quote}
There is a kind of history which depends altogether upon the testimony
of authorities\dots it is not really history at all, but we have no
other name for it\dots  History constructed by excerpting and
combining the testimonies of different authorities I call
scissors-and-paste history. 
\end{quote}
By contrast, the scientific historian will pay
attention to the latest research \cite[p.35]{Collingwood-PH}:
\begin{quote}
\dots whereas the books mentioned in a bibliography for use of a
scissors-and-paste historian will be, roughly speaking, valuable in
direct proportion to their antiquity, those mentioned in a
bibliography for the use of a scientific historian will be, roughly
speaking, valuable in direct proportion to their newness. 
\end{quote}
What this means for math history, I
think, is that we must not treat Euclid's \emph{Elements,} say, as
the word of God or even the unaltered word of Euclid.  We may well pay
attention to Russo's argument in `The First Few Definitions in the
\emph{Elements}' \cite[10.15, pp.~320--7]{MR2038833} that the obscure
definition of straight line now found in the \emph{Elements} is the
work, not of Euclid, but of a careless copyist.  Still, there is
little point in reading Russo without reading the text associated with
Euclid's name. 

\minisec{Experience}

Most of our students will not be professional mathematicians.  The experience of
making sense of a difficult text, getting up in front of an audience,
and talking about their understanding, will be more useful to our students
than any particular piece of mathematical knowledge.  Indeed, I think this is
so, even for the students who \emph{will} be mathematicians.  At any
rate, as I said, my own undergraduate education consisted entirely of
this kind of learning.  Any ability I have now as a teacher was
nurtured by this experience. 

\minisec{Tradition}

Many people derive satisfaction from their membership in a group.  The
group might be a political party, a nation, humanity, or the
supporters of a football team.  If one is studying mathematics, I
suppose the best group to feel oneself a member of is the group of
\emph{mathematicians,} if not just the group of \emph{thinkers.} 
By actually \emph{reading} Euclid and his successors, we come to know that we
are part of a tradition that dates back thousands of 
years.  This point is reinforced when we consider that much of the
mathematics that our 
undergraduates learn was created by mathematicians who had read
Euclid.  Most of our course Elementary Number Theory I (Math 365), for
example, can be found in Gauss's \emph{Disquisitiones Arithmeticae}
(1801), of which
Wikipedia\footnote{\url{http://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae},
  accessed June 19, 2010.} says: 
\begin{quote}
The logical structure of the \emph{Disquisitiones} (theorem statement
followed by proof, followed by corollaries) set a standard for later
texts.  
\end{quote}
This claim is not sourced, but it seems short-sighted: the
statement--proof, statement--proof style of mathematical writing is
found in Euclid, whom Gauss implicitly credits in his preface
\cite[p.~xvii]{Gauss}: 
\begin{quote}
Included under the heading ``Higher Arithmetic'' are those topics
which Euclid treated in Book VIIff.\ with the elegance and rigor
customary among the ancients\dots\footnote{The continuation of the
  sentence is, `but they are limited to the rudiments of the science.'
  There has indeed been progress since Euclid.} 
\end{quote}
We may not expect our students to write as well as Gauss, even if he
was only their age when he was writing; but they would do well to have
Euclid as a model (and Gauss).\footnote{In my experience, the best
  mathematical writers among our students at METU grew up in the
  former Soviet Union.  I don't know if something in the Soviet
  tradition should be credited.  On p.~\pageref{Soviet} I quote a
  Soviet textbook that I used in high school.} 

\minisec{Changes}

Even though mathematics has an age-old tradition, the subject has
changed since Euclid; but this can be difficult to 
see.  Obviously we have more theorems now; less obviously, the
\emph{spirit} of mathematics has changed.
In \emph{The Foundations of Geometry} \cite{MR0116216}, David Hilbert
appears to think that, in 
axiomatizing geometry, he is only refining the work of Euclid.  If
so, Hilbert is wrong.  We think today that Euclid's five postulates
are not in fact sufficient to justify all of his propositions; rather,
there are hidden assumptions, overlooked by Euclid, which Hilbert
uncovers.  Even in Proposition 1 of Book I of Euclid's
\emph{Elements,} there is an implicit assumption that two
circumferences, each containing the center of the other, must
intersect; this assumption is justified by no postulate.  But we have
no 
reason to think that Euclid is \emph{trying} to uncover all of his
`hidden assumptions'.  He is just writing down what is
true.  

One may say further the intersection of the circles in Euclid's I.1 is not a
\emph{hidden} assumption; the intersection is evident from the
diagram.  Today we think nonetheless that the existence the point of
intersection should still be noted separately in words or
symbols.  Evidently Euclid did not think the same way.
(See pp.~\pageref{one-one} and~\pageref{axioms} below.)  

Euclid also has no
notion of `non-Euclidean' geometry, so he has no need 
to distinguish his geometry logically from any other.
His postulates, along with his demonstrations, serve as a sort of
\emph{explanation} of why his propositions are true; but there is no
reason to expect the postulates and the demonstrations to provide a
\emph{complete} explanation,---if the notion of completeness even
makes sense in this context.  
Now, although he does not seem to say so clearly, it may well be that
\emph{Hilbert's} goal was a complete set of axioms for
geometry; but what could this have meant?  Since Hilbert's
\emph{Foundations,} several different 
notions of logical completeness have been defined.  Hilbert in fact
succeeded in writing down a \emph{categorical} set of axioms,
in that any two geometries in which the axioms are true
must be isomorphic to one another.  But we can hardly say that Euclid
aimed to do the 
same, if for him there was only one geometry.

At the beginning of `On teaching mathematics',
V. I. Arnold says:\footnote{I take the text
  from \url{http://pauli.uni-muenster.de/~munsteg/arnold.html} (July 30,
  2010), where the following explanation is given: `This is an
  extended text of the address at the discussion on teaching of
  mathematics in Palais de D\'ecouverte in Paris on 7 March 1997.}  
  \begin{quotation}
    Mathematics is a part of physics. Physics is an experimental
    science, a part of natural science. Mathematics is the part of
    physics where experiments are cheap. 

The Jacobi identity (which forces the heights of a triangle to cross
at one point) is an experimental fact in the same way as that the
Earth is round (that is, homeomorphic to a ball). But it can be
discovered with less expense. 

In the middle of the twentieth century it was attempted to divide
physics and mathematics.  The consequences turned out to be
catastrophic. Whole generations of mathematicians grew up without
knowing half of their science and, of course, in total ignorance of
any other sciences. They first began teaching their ugly scholastic
pseudo-mathematics to their students, then to schoolchildren
(forgetting Hardy's warning that ugly mathematics has no permanent
place under the Sun).  
  \end{quotation}
I can't say that Arnold is right about mathematics in general.  He
may be right to say that \emph{Euclid's} mathematics is physics (as
physics is understood today).  Again, Euclid \emph{explains} why many
geometric propositions are true.  If one finds the explanations
inadequate, one may probe further; this does not make Euclid wrong.
Likewise, we have many explanations
of the motions of the heavens---explanations by Ptolemy, Copernicus,
Kepler, Newton, and Einstein.  None of these explanations is wrong.
Each new explanation builds on the preceding, as Hilbert built on
Euclid; on the other hand, each physicist had a different project:
each was looking for a different \emph{kind} of answer to the question,
`Why do the heavens appear as they do?'\footnote{For example, Ptolemy
  wanted to know what configurations of circular
  motions could account for the dance of the planets in the sky.  From
  Kepler, Newton understood 
  that the planets moved in ellipses about the sun; Newton sought a
  different kind of account of this, namely a law of force.}

In the preceding paragraphs, I have expressed opinions about Euclid,
Hilbert, and others.  I might 
express these opinions to students; but the students should question the
opinion while consulting Euclid himself (and 
Hilbert, and the others).  It may well be that a modern mathematician
misunderstands his ancient predecessors, because his main business is
to be a mathematician and not an historian.  If one just wants to
learn mathematics from the mathematician, that's fine; if one also
wants to learn history, one should go to the source.

\minisec{Proof}

Many of Euclid's propositions are propositions that I learned to prove
in high school, albeit from a modern textbook.\footnote{I didn't much like
the textbook.  I wanted to read Euclid, and did so, first on my own,
and then at St John's.}  As I understood it, the
purpose of my high-school course was not so much to learn those
geometrical results themselves; the main purpose was to learn the
\emph{possibility} of proving those results.  Unfortunately our
students at METU seem never to get such a course, either in high
school or with us (see p.~\pageref{proofs-new}).  We do teach proof
here; but at the same time we are teaching modern mathematics, and
this complicates
things.  I count Descartes as modern.  Descartes gives us a
method of great power, which we start passing on to our students in
their first semester, in
Analytic Geometry (Math 115); but it is difficult to understand 
the method's power of \emph{proof.}\footnote{See
  \S~\ref{sect:exam-I-1} for an exam that required application of
  Descartes's analytic geometry, as well as Newton's conception of
  quadrature.  Most students performed very poorly on this exam; later
  I discuss what to do about this.} 

Using analytic methods, how would we prove that the
base angles of an isosceles triangle are equal?  Given such a
triangle, we can set up a rectangular coordinate system in
which the vertices $A$, $B$, and $C$ of the triangle are respectively
$(0,a)$, $(b,0)$, and $(c,0)$, where $b\neq c$; then $AB=AC$ if and only if
$a^2+b^2=a^2+c^2$, that is, $b=-c$ (since $b\neq c$).  In this case, the angles
at $B$ and $C$ 
have the same cosine, namely $\lvert b\rvert/\sqrt{a^2+b^2}$, so the angles
are equal.  Fine; but this argument uses notions not found till page
133 of the analytic geometry text~\cite{Karakas} used at METU; even then, the text just assumes familiarity with cosines, when full
knowledge of these will not come till a later course of mathematical analysis.

By contrast, for Euclid, the equality of the base angles of an
isosceles triangle is only Proposition 5 of Book I of the
thirteen books that make up the \emph{Elements.}  Notwithstanding the `hidden assumptions' mentioned above,
I don't know anything better than Euclid's `synthetic' geometry for
giving students a notion of what is a sound proof.

\minisec{Thrills}

I just mentioned the power of Descartes's analytic geometry.  It is a
thrill to learn this geometry from Descartes himself.  The thrill is
worth sharing with students; but it does not come 
cheap (comments on p.~\pageref{fun} notwithstanding).  One needs to
have read Descartes's predecessors, and to have
read them faithfully---\emph{not} translated into modern, symbolic,
\emph{Cartesian} language.  But textbooks like Boyer \cite{MR996888} present
the old work in just this anachronistic way.

\minisec{Discoveries}

It can happen that new mathematics comes out of taking old mathematics
seriously.  I can offer only my own example.  One of my papers, about the
logic of vector spaces \cite{MR2505433},\footnote{The main 
  mathematical result is that if we have a vector space of dimension
  greater than $n$, then we can enlarge the scalar field so that the
  dimension of the space is reduced to $n$, while every set of $n$
  vectors that are linearly independent over the original scalar
  field remain independent over the new field.  I quote from the anonymous referee:
  \begin{quote}
The paper is well-written and very interesting. The structures are
indeed basic, yet I found several results which surprised me, and the
technical profficieny with which things are handled makes publishing
the presentation worth-while. For example realizing the geometric idea
of Descartes, while taking care to make all formulae existential, is
an example of the added value of the paper. To me personally even the
fact that the scalar field can be recovered from the parallelism
predicate was new.
  \end{quote}} is directly
inspired by reading Euclid and Descartes.

\minisec{Coverage}

The wonderful new \emph{Princeton Companion to Mathematics}~\cite{MR2467561} contains short biographies of 96 mathematicians, in chronological order.  The first 14 mathematicians listed are:
\begin{compactenum}[(1)]
\item
Pythagoras,
\item
Euclid,
\item
Archimedes,
\item
Apollonius,
\item
Abu Ja'far Muhammad ibn M\=us\=a al-Khw\=arizm\=\i,
\item
Leonardo of Pisa (known as Fibonacci),
\item
Girolamo Cardano,
\item
Rafael Bombelli,
\item
Fran\c cois Vi\`ete,
\item
Simon Stevin,
\item
Ren\'e Descartes,
\item
Pierre Fermat,
\item
Blaise Pascal,
\item
Isaac Newton.
\end{compactenum}
In my course, we read works of seven of the mathematicians on this
list.  (There is no extant text by Pythagoras.  In one class I
lectured additionally on Archimedes.) 
We also read two
other mathematicians, namely Th\=abit ibn Qurra and Omar Khayy\'am;
but we could have dropped the former, in line with the suggestion of Ali in
\S~\ref{sect:comments-spring}.  It is indeed a shame not to read any
of the remaining 82 mathematicians on the \emph{Princeton Companion's}
list.  But there just isn't time to read many more.  One could read
the work of many mathematicians in a source book like
Smith's~\cite{MR0106139} or Struik's~\cite{MR858706}, but I think the
coverage would be too superficial to be of much value. 

If it is desired, Newton's contemporaries (such as Leibniz) and
successors can be studied in the courses that 
cover their work.  About courses I have taught, I can say that Gauss
can be read in Math 365, while Set Theory (Math 320) and Introduction
to Mathematical Logic and Model Theory (Math 406) can make use of van
Heijenoort's anthology~\cite{MR1890980}.  It just does not seem fair
to me to use a course like Math 303--304 to teach students about
mathematicians whose work they do not have time to \emph{know.}   

If one wants a royal road to a view of the grand sweep of mathematical
history, one can read Struik's \emph{Concise History of Mathematics}
\cite{Struik}.  However, I am uneasy with Struik's materialistic
approach.  He writes for example: 
\begin{quote}
The rapid development of mathematics during the Renaissance was due
not only to the \emph{Rechenhaftigkeit}\footnote{\emph{Calculability,}
  according to
  \url{http://www.dict.cc/german-english/Rechenhaftigkeit.html}
  (accessed June 23, 2010).} 
of the commercial classes but
also to the productive use and further perfection of machines. 
\end{quote}
This is an explanation, perhaps correct; but it is hardly complete.
The development of mathematics is, first of all, due to
\emph{mathematicians.}  This is a point worth making in a course, and
it is a point made by the practice of reading \emph{mathematicians.} 
Struik may not disagree.  In the introduction of his history, he writes:
\begin{quote}
  The selection of material was, of course,  not based exclusively on
  objective factors, but was influenced by the author's likes and
  dislikes, his knowledge and his ignorance.  As to his ignorance, it
  was not always possible to consult all sources first-hand; too
  often, second- or even third-hand sources had to be used.  It is
  therefore good advice, not only with respect to this book, but with
  respect to all such histories, to check the statements as much as
  possible with the original sources.  This is a good principle for
  more than one reason.  Our knowledge of authors such as Euclid,
  Diophantus, Descartes, Laplace, Gauss, or Riemann should not be
  obtained exclusively from quotations or histories describing their
  works.  There is the same invigorating power in the original Euclid
  or Gauss as there is in the original Shakespeare, and there are
  places in Archimedes, in Fermat, or in Jacobi which are as beautiful
  as Horace or Emerson.
\end{quote}

\addsec{Possibilities for the future}

I would make some changes in teaching Math 303--304 again.  Here are
some notes about what might be done. 

If students are going to make presentations, they must prepare for these
conscientiously, with the understanding that a poor presentation will
disappoint not only their teacher.  \emph{Classmates} must challenge
students who try to fake their way through a proof.  Such challenges
happened occasionally in my class (see for example
p.~\pageref{confused}); I wish I could encourage students 
to make more of them, or (better) convince the speakers not to try to
fake their way.  (See pp.~\pageref{scolding}, \pageref{Besmir-silent},
and~\pageref{Melis-copying}, and \S~\ref{sect:frust} 
[p.~\pageref{Emir-reading}], for some problematic days.)  

Some 
formal measures might be of help.  I did learn all 
of my students' names; but I found out too late that they didn't
always know \emph{one another's} names.  I sometimes tried arranging
the desks in a semicircle (see pp.~\pageref{desks}
and~\pageref{desks-again}).  I am told by Mehmet (whom I mentioned
above) that what I really must do is \emph{grade} the students on
their individual presentations.  Mehmet is not a student who needs
such a goad, but (if I understand him) other students do need prodding
by the threat of low marks.  In this case, I can only
hope that what students first do for marks, they may later do for
their own satisfaction.  As it was, I did tell students that they got
credit for \emph{attending} class; I did not say that students would
be graded on the \emph{quality} of their attendence and participation.

  Also (suggests Mehmet), students should know many weeks in advance
  what they will be presenting.  This should be possible, now that I
  know (from this very log) at what pace the course can proceed.
  Mehmet did think the practice of reading original sources like Newton
  should be continued.\footnote{Mehmet took both semesters
    of Math 303--304 and is now going to study for a doctorate in
    physics at Yale.}) 

Classes proceeded more slowly than I expected, sometimes because
students had indeed not conscientiously prepared for them.  If one wants to
cover more material, one can skip some 
propositions in class, while holding the students responsible for
learning them independently.  Students might still work together, as Ece
suggested; see \S~\ref{sect:comments-spring}.  Still, it should be
noted that, though at the beginning of Math 304 I assigned
presentations to pairs or triples of students, the students
generally didn't work together.

The teacher could compromise his principles and make 
some presentations himself.  Indeed, as I noted above, I did this with
Archimedes.  I did it too with Book V of the 
\emph{Elements,} on proportion (see p.~\pageref{proportion}), and I
should have done it more; here, understanding the mathematics is hard 
enough, even if one is not trying to learn the mathematics straight
from Euclid.  The final exam of Math 303 showed that students had
\emph{not} generally learned Euclid's definition of proportion (the
one that must have inspired Dedekind's definition of the real numbers
\cite{MR0159773}).\footnote{Russo~\cite[pp.~46 f.]{MR2038833}
  ridicules historians like Heath, who are impressed that Euclid could
  have `anticipated' Dedekind's theory of irrational numbers.  Euclid
  didn't anticipate Dedekind; he \emph{taught} Dedekind, who read him
  in school.} 

It was hard for the students not to have much sense of what would be
on exams.  I didn't have much sense myself, when I started the course.
Nonetheless, in the first semester, students generally impressed me by
their understanding on exams; but in the second semester, they
disappointed me.  I was quite pleased with the problems I wrote on the
last two exams of Math 304; but in another year, some such problems
should be worked out with students in class.  

In lectures in the second semester, I compromised and \emph{stated}
for the students the results from Apollonius that we would need for
Newton.  Proofs of some of these results did however end up as the
exam problems just mentioned. 
Again, it would be better to make proofs of \emph{all} of these
results more clearly a part of the class, either in lectures or in
homework.  Proofs can use the streamlining that Descartes makes
possible, at least if the point is to be able to read Newton's
\emph{Principia.}  (In Math 304, I told the students out loud that
problems like those on the second exam could show up on the final; but
the students seemed not to do anything with this warning.) 

Anthologies like
Katz~\cite{Katz} are useful for identifying the old works of
mathematics that may be worth reading.  However, it may be misleading
to see a brief excerpt out of context.  It would be desirable (where
possible) to consider an anthology's selections in the context of the
larger works from which they have been taken.\footnote{I have
  therefore asked our library to order some of these larger works.}

Unfortunately most students of Math 304 probably will not have taken
Math 303.  Therefore it may be better to divide the contents of the
course not chronologically, but thematically, perhaps with geometry
and analysis in one semester, number theory and 
algebra in the other.  The former could start with Book I of Euclid's
\emph{Elements;} the latter could leave this book as background
reading, but start seriously with Book II. 

I had originally
thought of finishing Math 304 with Lobachevsky, but there was no
time, and anyway most of the students had not read Euclid, because
they had not taken Math 303.  In a
rearrangement of the course, Lobachevsky could be accommodated
somehow.  On the other hand, Lobachevsky is number 31 on the
(chronological) list in the \emph{Princeton Companion;} we would skip
a lot of great names to get to him.

\begin{flushright}
D. P.\\
Ankara\\
\today
\end{flushright}

%\addtocontents{toc}{\enlargethispage{1\baselineskip}}
%\tableofcontents

\part{Fall semester}\label{part:one}

\input{math-303-log}

\part{Spring semester}\label{part:two}

\input{math-304-log}

\appendix

\chapter{Examinations}\label{app:exams}

\input{history-exams}

\chapter{Student comments}\label{app:comments}

\input{student-comments}

\chapter{Collingwood on history}\label{app:Col}

\input{collingwood}

\chapter{Departmental correspondence}\label{app:cor}

\input{correspondence}

\chapter{Notes on Greek mathematics}\label{app:background}

\input{background}

% \bibliographystyle{amsplain}
% \bibliography{../../../../TeX/references}
 %\bibliography{../../references}

\def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{10}

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\end{document}
