\documentclass[% version=last,% b5paper,% %letterpaper,% 10pt,% %12pt,% headings=small,% bibliography=totoc,% index=totoc,% twoside,% reqno,% cleardoublepage=empty,% open=any,% draft=false,% DIV=classic,% %DIV=10,% headinclude=false,% pagesize]%{scrreprt} {scrbook} %\usepackage[notref,notcite]{showkeys} \setcounter{tocdepth}{1} \usepackage[OT2,T1]{fontenc} \usepackage[russian,polutonikogreek,english]{babel} \newcommand{\Ru}[1]{\foreignlanguage{russian}{#1}} %\usepackage{gfsneohellenic} \newcommand{\Gk}[1]{\foreignlanguage{polutonikogreek}{#1}} \newcommand{\Engl}[1]{\emph{#1}} \newcommand{\Tur}[1]{\textsf{#1}} \newcommand{\letter}[1]{\textsf{#1}} %for letters as such %\usepackage{wrapfig} \usepackage{scrpage2} \pagestyle{scrheadings} \ohead{\pagemark} \ihead{\headmark} \ofoot{} \renewcommand{\captionformat}{ } \usepackage{layout} \usepackage{amsmath,amsthm} \usepackage{mathrsfs} \usepackage{cclicenses} \usepackage[perpage,symbol]{footmisc} %\renewcommand{\thefootnote}{\fnsymbol{footnote}} \usepackage{wasysym} % for \smiley \usepackage{url} \usepackage{amssymb} \usepackage{hfoldsty} \usepackage{pstricks,pst-node,pst-tree,pst-plot} \usepackage[neverdecrease]{paralist} \usepackage{graphicx} \usepackage[all]{xy} \usepackage{verbatim} \usepackage{amscd} \usepackage{upgreek} \newcommand{\comprat}{\mathbin{\&}} \newcommand{\asterism}{\mbox{}\hfill\hfill$*$\hfill$*$\hfill$*$\hfill$*$\hfill$*$\hfill\hfill\mbox{}} \newcommand{\rect}[1]{\operatorname{rect.}#1} \newcommand{\sq}[1]{\operatorname{sq.}#1} %\newcommand{\ngt}{\mathop{\tilde{\mathrm m}.}} \newcommand{\mns}{\mathbin{\tilde{\mathrm m}.}} \newcommand{\pls}{\mathbin{\tilde{\mathrm p}.}} %\newcommand{\Rx}{\mathop{\mathrm{P\!\!_X}}} \usepackage{textcomp} \newcommand{\Rx}{\mathop{\text{\textrecipe}}} \newcommand{\mi}{\mathrm i} \newcommand{\R}{\mathbb R} \newcommand{\size}[1]{\lvert#1\rvert} \newcommand{\ult}{\operatorname{ult}} \newcommand{\arc}{\operatorname{arc}} \renewcommand{\leq}{\leqslant} \renewcommand{\geq}{\geqslant} %\renewcommand{\phi}{\varphi} \newcommand{\measures}{\mathrel{\mid}} \newcommand{\ratio}[2]{#1:#2} \newcommand{\psign}{\mathrel{:\,:}} \newcommand{\eqr}[2]{#1\psign#2} \newcommand{\prop}[4]{\eqr{\ratio{#1}{#2}}{\ratio{#3}{#4}}} \newcommand{\equim}[4]{\eqr{#1\measures#2}{#3\measures#4}} \renewcommand{\land}{\mathrel{\&}} \renewcommand{\theequation}{\fnsymbol{equation}} \newtheorem{problem}{Problem}[section] \newtheorem*{bonus}{Bonus} \newtheorem*{enunciation}{Enunciation} \theoremstyle{definition} %\newtheorem{problem}{Problem} \newtheorem*{solution}{Solution} \newtheorem{proposition}{Proposition} \theoremstyle{remark} \newtheorem*{remark}{Remark} \newtheorem*{exposition}{Exposition} \newtheorem*{specification}{Specification} \newtheorem*{construction}{Construction} \newtheorem*{conclusion}{Conclusion} \renewcommand{\arraystretch}{2} %\renewcommand{\labelenumi}{\theenumi.} \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, 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} \bibitem{MR0053831} {Apollonius of Perga}, \emph{On conic sections}, Great Books of the Western World, no. 11, Encyclopaedia Britannica, Inc., Chicago, London, Toronto, 1952. \MR{MR0053831 (14,832c)} \bibitem{MR1053158} \bysame, \emph{Conics. {B}ooks {V}--{VII}}, Sources in the History of Mathematics and Physical Sciences, vol.~9, Springer-Verlag, New York, 1990, The Arabic translation of the lost Greek original in the version of the Ban\=u M\=us\=a, Volume I: Introduction, text and translation, Volume II: Commentary, figures and indexes, Edited with translation and commentary by G. 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