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\title[Returning to the Roots]{Returning to the Roots of Mathematics:\\ 
A Personal Journey\\
$ $\\
                               Faculty Scholarship Lunch}
\author
{John B. Little}

\institute[Math and CS]{
  Department of Mathematics and Computer Science
}

\date[February 14, 2017]
{February 14, 2017}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%{\bf Outline}

%\tableofcontents

%\end{frame}

\section{Introduction}

\begin{frame}
{\bf A true story}

Soon after I started at Holy Cross, former Science Librarian Tony Stankus pointed out an extraordinary
``flame war'' carried out mid-1970's in {\it Arch. for Hist. of Exact Sci.} A sample:
\vskip 5pt
{\it `` ... history of mathematics has been typically written by mathematicians ... who have either reached 
retirement age and ceased to be productive in their own specialties or become otherwise professionally sterile.
... The reader may judge for himself how wise a decision it is for a professional to start writing the history 
of his discipline when his only calling lies in professional senility.''}
\vskip 5pt 
I agreed(!)  Wonderful irony and/or a complete confirmation of the author's point: 
I've recently gotten interested in the specific issues this particular article dealt with(!)  More on that later ... 
\end{frame}

\begin{frame}
{\bf You can probably sympathize}
\begin{itemize}
\item<1-> That awkward question ... ``So, what do you do?''
\item<2-> Because I'm an \emph{algebraic geometer} by training, often
start out by saying, ``Well, I'm a mathematician -- I study geometric objects defined by certain
kinds of algebraic equations, ... ''
\item<3-> ``For instance, the \emph{conic sections} (ellipses, parabolas, hyperbolas)
all defined by second degree polynomial equations in two variables: 
$$A x^2 + B xy + C y^2 + D x + E y + F = 0$$
for some coefficients $A,B,C,D,E,F$.''
\item<4-> ``It all goes back to the ancient Greeks ... ''
\end{itemize}
\end{frame}

\begin{frame}
{\bf My own changing interests}

\begin{itemize}
\item Of course, in this form, this uses
key insights of {\it Ren\'e Descartes} leading to {\it analytic} 
(i.e. coordinate) geometry -- the Greeks did things differently.
\item But at some point, I started to wonder: \emph{Exactly how did} ancient Greek mathematicians such as 
Apollonius really think about the conic sections?  
\item Return to the roots! Goal:  Read Apollonius (and Descartes) but not filtered through 
modern "interpretations," "explanations" and translations
\item I had to learn some ancient Greek ... lucky I'm at Holy Cross with our world-class 
and wonderfully welcoming Classics department!
\end{itemize}

\end{frame}

\section{How Apollonius described and classified the conic sections}


\begin{frame}
{\bf Context -- previous work on conics}

\begin{columns}
\column{.5 \textwidth}
\begin{figure}
\includegraphics[height=2.0in, width=1.8in]{orthotome.png}
\end{figure}
(Image by H. Mendel, Cal. State LA)

\column{.45 \textwidth}
\begin{itemize}
\item Menaechmus (ca. 380 - 320 BCE; in Plato's circle) often credited with discovery
\item Developed by Aristaeus (pre-Euclid) and Euclid (ca. 300 BCE) 
\item Known only through later commentaries and works by Archimedes; e.g. 
the parabola as a ``section of a right-angled cone''
\end{itemize}
\end{columns}
\end{frame}

\begin{frame}
{\bf Apollonius of Perga, ca. 262--190 BCE}

\begin{itemize}
\item<1-> Active roughly 75 - 100 years after time of Euclid (ca. 300 BCE); 
slightly younger than Archimedes (ca. 287--212 BCE); studied in Alexandria
\item<2-> Have lists of his works from later commentaries, but most have not survived
\item<3-> The \emph{Conics}:  Books I, II, III, IV -- ``Elements of conics'' -- survive in Greek versions. 
\item<4-> Book V, VI, VII -- ``Researches on conics'' -- only known in Arabic
\item<5-> Book VIII -- ? (lost -- several attempts at ``reconstruction'' including one by E. Halley, 1710 CE)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Apollonius's framework}

\begin{columns}
\column{.5 \textwidth}
\begin{figure}
\includegraphics[height=2.0in, width=1.8in]{conics.png}
\end{figure}
(Image from Wikipedia, ``Conic Sections'')

\column{.45 \textwidth}
\begin{itemize}
\item Conic surfaces generated by lines through a vertex and points on a circle
\item All conics are obtained by slicing any one conic surface
by different planes.
\item Get plane curves, but the construction inherently uses geometry in 3 dimensions(!)
\end{itemize}
\end{columns}

\end{frame}


%\begin{frame}
%{\bf {\it Sumptomata} of conics}
%
%\begin{columns}
%\column{.5 \textwidth}
%\begin{figure}
%\includegraphics[height=1.6in, width=1.3in]{ApPropI11.pdf}
%\end{figure}
%(Image from Apollonius, {\it Conics}, trans. R.C.Taliaferro, Green Lion Press)
%\column{.45 \textwidth}
%\begin{itemize}
%\item<1-> Apollonius classifies the conic sections according to the way the cone
%is sectioned
%\item<2-> The ``{\it sumptomata}'' are expressed in each case as
%a relation between a given square and a given rectangle 
%\end{itemize}
%\end{columns}
%\end{frame}

\begin{frame}
{\bf Book I Proposition 11 -- definition of the parabola (R.C.Taliaferro's translation)}

\SmallFont
If a cone is cut by a plane through its axis, and also cut by another plane cutting the base of the
cone in a straight line perpendicular to the base of the axial triangle, and if, further, the diameter
of the section is parallel to one side of the axial triangle, and if any straight line is drawn from 
the section of the cone to its diameter such that this straight line is parallel to the common section of 
the cutting plane and of the cone's base, then this straight line to the diameter will equal in square 
the rectangle contained by (a) the straight line from the section's vertex to where the straight line
to the diameter cuts it off and (b) another straight line which has the same ratio to the straight 
line between the angle of the cone and the vertex of the section as the square on the base of the 
axial triangle has to the rectangle contained by the remaining two sides of the triangle.  And let
such a section be called a parabola. 
\end{frame}

\begin{frame}
{\bf Comments}
\begin{itemize}
\item<1->Apollonius' prose is verbose, complicated in syntax, {\it and mathematically dense} -- 
a ``hard slog!''
\item<2->  J. Kepler:  
``If anyone thinks that the obscurity of this presentation arises from the perplexity of my mind,
... I urge any such person to read the {\it Conics} of Apollonius.  He will see that there are some
matters which no mind, however gifted, can present in such a way as to be understood in a cursory 
reading.  There is need of meditation, and a close thinking through of what is said.''
\item<3-> Descartes, from \emph{La G\'eom\'etrie}:
``\emph{ ... je vous prie de remarquer en passant que le scrupule que faisoient les
anciens d'user des termes de l'arithm\'etique en la g\'eom\'etrie ... 
%qui ne pouvoit
%proc\'eder que de ce qu'ils ne voyoient pas assez clairement leur rapport, 
causoit beaucoup d'obscurit\'e et d'embarras en la fa\c{c}on dont ils s'expliquoient ... }'''
\end{itemize}
\end{frame}

\begin{frame}
{\bf Structure of a Euclidean or Apollonian proposition}

\begin{itemize}
\item<1-> Typically, there are 6 ``parts'' -- {\it protasis}, [diagram and] {\it ekthesis}, {\it diorismos}, 
{\it kataskeu$\overline{e}$}, {\it apodeixis}, {\it sumperasma}
\item<2-> The supremely complex and convoluted first sentence above is the {\it protasis} of this
proposition -- the ``statement''
\item<3->  The {\it ekthesis} then ``lays out'' the statement by means of a figure and the
sort of labeling of important points with letters familiar from school geometry
(a \emph{lifesaver} for the reader!)
\end{itemize}
\end{frame}


\begin{frame}
{\bf {\it Ekthesis} of Proposition 11, beginning (my condensed translation)}

\begin{columns}
\column{.5 \textwidth}
\begin{figure}
\includegraphics[height=1.8in, width=1.4in]{ApPropI11.pdf}
\end{figure}
\column{.45 \textwidth}
Let $A$ be the vertex, $\Delta ABC$ the axial triangle, 
and let the other plane cut the plane of the base
in $DE$ perpendicular to $BC$ so that $FG$ is \emph{parallel to} $AC$.  
The section is the curve $DFE$.
\end{columns}
\end{frame}

\begin{frame}
{\bf Conclusion of {\it ekthesis} and {\it diorismos} of Proposition 11}
\begin{columns}
\column{.35 \textwidth}
\begin{figure}
\includegraphics[height=2.5in,width=1.8in]{ApPropI11.pdf}
\end{figure}
\column{.60 \textwidth}
Let $H$ be ``contrived so that'' 
$$sq. BC : rect. BA,AC :: FH : FA\leqno(*)$$
Finally let $K$ be taken at random on the section
and let $KL$ be parallel to $DE$.
\vskip 5pt
I say that $sq. KL = rect. HF,FL$.  
\end{columns}
\end{frame}

\begin{frame}
{\bf A word on terminology and notation}
\begin{itemize}
\item The notation here is Taliaferro's modern attempt to capture Apollonius 
in a (more) readable way
\item The Greek is highly conventionalized and abbreviated
\item Here $sq. XY$ means (the area of) the square with side $XY$ (Apollonius just
says \emph{to apo XY}: literally ``the from XY'')
\item $rect. XY,YZ$ stands for (the area of) the rectangle with sides $XY$ and $YZ$
(\emph{to upo XYZ}: literally  ``the by XYZ'')
\item The $:$ and $::$ are standard notation for comparing proportions that may be
familiar from analogies
%(Books V, X of Euclid
%contain an exposition of Eudoxus' theory of these)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Historical comments}

\begin{itemize}
\item<1->  This fact about parabolas was \emph{certainly not discovered by Apollonius} 
\item<2->  Archimedes' \emph{Quadrature of the Parabola}, for instance, 
states something very close, precisely -- if $KL$ and $K'L'$ are \emph{two} such segments, then 
$$sq. KL : sq. K'L' :: FL : FL',$$ 
and it's given \emph{without proof}.
Usual interpretation (almost certainly correct): Archimedes is relying on a standard
reference, probably the lost Euclid {\it Conics}
\item<3->  What seems to be new in Apollonius is the use of this property to \emph{define}
a parabola, \emph{and} the names we use for this curve and the other conic sections
\end{itemize}
\end{frame}

\begin{frame}
{\bf The names ``parabola,'' ``hyperbola'' and ``ellipse''}

\begin{itemize}
\item<1->  Greek mathematical terminology often ``borrowed''
common words and gave them special meanings.
\item Apollonius did this for the conics
 (following earlier work in a different context -- ``application of areas'')
\item {\it parabol$\overline{e}$} -- noun: a ``throwing alongside,'' comparison, juxtaposition
\item {\it hyperbol$\overline{e}$} -- noun: a ``throwing beyond,'' excess, superiority 
\item {\it elleip$\overline{o}$} -- verb:  to fall short, be in want of, lack 
\end{itemize}
\end{frame}

\section{Translation and historiography of mathematics}

\begin{frame}
{\bf What's in a word?}

\begin{itemize}
\item<1-> The Greek phrase Apollonius 
uses for the segments $KL$ before deserves special consideration:
``\emph{tetagmen$\overline{o}$s} to the diameter'' -- 
literally means something like \emph{``lined-up''} or \emph{``in order''}, or perhaps even 
``drawn \emph{in an orderly fashion}'' (from the section to its diameter)
\item<2-> OTOH, many standard English translations of Apollonius (e.g. Heath, Taliaferro, ... ) 
say those parallels have been drawn \emph{``ordinatewise''}  to the diameter.
\item<3->  Interestingly enough, the entry in the standard LSJ Greek lexicon for 
\emph{tetagmen$\overline{o}$s} gives the common meaning and then ``ordinatewise'' with a reference
to Definition 4 in Book I of Apollonius(!) My guess: some mathematical historian 
{\it (maybe T.L.Heath?)} provided this citation to the compilers of the lexicon(!)
\end{itemize}
\end{frame}

\begin{frame}
{\bf \emph{tetagmen$\overline{o}$s} to ``ordinatewise?''}

\begin{itemize}
\item First Latin translation of Apollonius to 
circulate widely in Western Europe by Federigo Commandino (1509-1575 CE);
then several others too, including one by E. Halley (1656-1742 CE).
\item Commandino's Latin rendering: \emph{ordinatim applicatae} -- pretty literal version of the 
\emph{everyday Greek meaning of the word -- ``applied in an orderly fashion.''}  
\item Halley (also Latin) has something equivalent; later uses the word {\it abscissae} for 
distances along the segments ``cut off'' by the diameter and the section. 
\item Note slightly old-fashioned analytic geometry terminology: 
``abscissas and ordinates'' are $x$ and $y$ coordinates(!)  
Was Apollonius was thinking in coordinate terms after all?? 
\end{itemize}
\end{frame}

\begin{frame}
{\bf Apollonius' ``equation of a parabola?''}

\begin{itemize}
\item<1-> Recall the {\it diorismos} of Apollonius' Proposition 11:  ``I say that $sq. KL = rect. HF,FL$.''  
\item<2-> Segments like $KL$ are said to be drawn ``ordinatewise'' (mis?)reading Commandino
\item<3-> If we write the ``ordinates'' $y = \overline{KL}$ and ``abscissas'' $x = \overline{FL}$, 
then noting that $HF$ is a fixed
segment of length $c$, say, we get the ``sideways'' parabola $y^2 = c x$ (and $c$ corresponds to 
the length $HF$ -- called the \emph{orthia pleura} or ``upright side'' in Apollonius -- ``latus rectum''
later) 
\item<4-> Can get analogous statements for the hyperbola and ellipse as well!
%\item<5-> Moreover, the whole proof can be rephrased as manipulations of \emph{fractions} with numerator and
%denominator equal to lengths of segments (good exercise; Heath's version of Apollonius
%does this consistently)
\end{itemize}
\end{frame}

%\begin{frame}
%{\bf It's almost too hard not to do it ... }
%
%\begin{itemize}
%
%\vskip 5pt
%``\emph{ ... je vous prie de remarquer en passant que le scrupule que faisoient les
%anciens d'user des termes de l'arithm\'etique en la g\'eom\'etrie, qui ne pouvoit
%proc\'eder que de ce qu'ils ne voyoient pas assez clairement leur rapport, 
%causoit beaucoup d'obscurit\'e et d'embarras en la fa\c{c}on dont ils s'expliquoient ... }''
%\end{itemize}
%\end{frame}

\begin{frame}
{\bf Apollonius ``interpreted'' for modern mathematicians}

\begin{itemize}
\item<1-> H. Zeuthen, {\it Die Lehre von den Kegelschnitten im Altertum}, first ed. 1886 
\item<2-> T.L. Heath's version of Apollonius, first ed. 1896 -- 
``\emph{so entirely remodelled by the aid of accepted modern notation as to be thoroughly 
readable by any competent mathematician}''  since
it ``\emph{does not essentially differ from ... modern analytic geometry except that in 
Apollonius geometrical operations take the place of algebraical calculations}''
\item<4-> C. Boyer (1906-1976)
``\emph{The work of Apollonius in many respects approaches so closely to the modern
form of treatment that it not infrequently has been regarded as constituting analytic geometry}.''
\end{itemize}
\end{frame}

\begin{frame}
{\bf A contrary view}

\begin{itemize}
\item S. Unguru (1931-present), ``On the need to rewrite the history of Greek mathematics,'' 
Archive for History of Exact Sciences 15 (1975/76), 67--114 (the article my introductory quotation came from!)
\item Unguru later forcefully refutes prevailing mathematical historiography of the era of Zeuthen
and Heath and the 
use of algebraic reformulations to explain Apollonius in a 2001 book with M. Fried
\item Unguru's main point: it's geometry pure and simple; Greek mathematics did not have any of the apparatus of symbolic algebra or coordinates
\end{itemize}
\end{frame}

%\begin{frame}
%{\bf Heath ``for the defense''}
%
%\begin{itemize}
%\item It should be Apollonius and nothing but Apollonius, and nothing should be altered in the substance
%or in the order of his thought
%\item It should be complete, leaving out nothing of any significance or importance
%\item It should exhibit under different headings the successive divisions of the subject, so 
%that the definite scheme followed by the author may be seen as a whole
%\item Apollonius' method 
%``\emph{does not essentially differ from that of modern analytic geometry except that in 
%Apollonius geometrical operations take the place of algebraical calculations}''
%\end{itemize}
%\end{frame}

\begin{frame}
{\bf Unguru's point, expanded}

\begin{itemize}
\item<1-> ``Explaining'' Apollonius this way (and other similar ``reconstructions''
of Greek mathematics using modern concepts from the 19th and early 20th centuries) is {\it perniciously wrong} from the historical point of view 
\item<2-> A false description of a fundamentally different understanding of mathematics 
\item<3-> {\it Conceptual anachronism} or ``Whig history'' -- presents the past as leading inevitably to the present 
\item<4-> Apollonius (following Euclid) {\it never} uses a numerical value as a measure of length 
or area -- coordinate equations ``don't compute'' in this context
\end{itemize}
\end{frame}

%\begin{frame}
%{\bf A ``flame war,'' part 1}
%
%\begin{itemize}
%\item When Unguru's 1975 article appeared, one of the mathematicians/historians 
%Unguru had savaged, B. L. van der Waerden, 
%was still alive (his dates: 1903--1996 CE).  
%\item You can imagine how well he liked that passage from Unguru's article!
%\item He published a rejoinder in the same issue of the Archive for History of Exact 
%Sciences -- ``A defence of a `shocking' point of view,'' 199--210.
%(The ``shocking'' was Unguru's characterization(!))
%\end{itemize}
%\end{frame}

%\begin{frame}
%{\bf ``Flame war,'' parts 2 and 3}
%
%{\it H. Freudenthal}: ``What is algebra and what has it been in history?'' Archive for History of Exact 
%Sciences 16 (1976/77), 
%189--200.  Argues that there is indeed algebra in Greek mathematics, using examples from Archimedes.
%But of course, a historian would say ``I thought we were talking about Euclid and/or Apollonius'' (quite 
%different)
%\vskip 5pt
%{\it A. Weil}: ``Who betrayed Euclid? Extract from a letter to the editor,''  Archive for History of Exact 
%Science 19 (1978/79), 
%91--93.  Essentially asks: ``who was responsible for allowing such a trashy, polemical article to be 
%published?  
%What is happening to the quality of this journal?''  And lands a blow:  ``... it is well to know 
%mathematics before concerning oneself with its history ... ''
%\end{frame} 

\begin{frame}
{\bf What can we learn from this?}

\begin{itemize}
\item<1-> May seem like a minor thing
\item<2-> But it points out a fundamental difference between doing mathematics and doing history of mathematics (as history)
\item<3-> Recognizing logical connections between old and new work and making reinterpretations is a part of what mathematicians do.
\item<4-> When apparently different things are logically the same, just expressed in different ways, 
mathematicians can and do treat them as 
the same(!)  And we are always looking for those equivalences--finding them can represent an advance in our understanding!
\end{itemize}
\end{frame}

\begin{frame}
{\bf And maybe Unguru had a (small) point?}

\begin{itemize}
\item<1-> As Unguru insinuated in his own nasty way, Zeuthen, his ``flame war opponents'' van der Waerden, Freudenthal, Weil, etc. were certainly all primarily mathematicians who had eminent research records 
and then turned to writing history later in their professional careers
\item<2-> Not surprising that they had the ``habits of mind'' and point of view of working mathematicians, not historians!
\item<3-> In particular, to put words in their mouths:  ``if it's logically equivalent to a coordinate
equation of a parabola, but expressed in geometric terms, 
then it's still essentially a coordinate equation''
\end{itemize}
\end{frame}

\begin{frame}
{\bf The ``take-home'' message}

\begin{itemize}
\item<1-> For intellectual historians, not so much logical equivalences that matter--it's particular features, differences!  Each culture, era, scientific school, etc. is a unique and separate thing
\item<3-> Unguru: The mathematical historian's first and most important job is to understand a body of mathematical 
work on its own terms, \emph{not on our terms}
\item<4-> A fundamentally different way of thinking
\item<5-> The title of a recent article by K. Saito:  ``Mathematical Reconstructions Out, 
Textual Studies In'' summarizes what's up in mathematical historiography these days!
\end{itemize}
\end{frame}

\begin{frame}
{\bf To be clear ... }

\begin{itemize}
\item<1-> In a recent article, ``Apollonius, Davidoff, Rorty, and Zeuthen: From A to Z, what else is there?'' (Sudhoffs Archiv, 91
(2007), 1 - 19), Unguru and Fried make it clearer that their ``issues'' concern Zeuthen's work 
{\it qua history}, not {\it qua} mathematics
\item<2-> and they contrast Zeuthen's well-intentioned and mathematically astute (mis)reading with a parodied ``post-modern,'' deconstructionist view that would deny {\it any} intrinsic meaning in a text
\item<3-> Make their point via a (hilarious, fictional) ``sexual politics reading'' 
of Apollonius. (Recall the {\it orthia pleura}?  It's {\it phallic}; you get the idea!)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Conclusions}

\begin{itemize}
\item<1-> The {\it Conics} is a masterwork (especially the later books), but
{\it it's not} coordinate geometry (that didn't exist yet and he \emph{never} 
uses coordinates on the whole plane!)  But, of course, 
his work (together with summaries, commentaries) \emph{was} read very 
carefully by Descartes and others and stimulated the development of analytic geometry(!)
\item<2-> I think the standard histories of mathematics still seriously misrepresent a 
lot of this and that does a disservice to teachers and students; I want to try to 
participate in an active way and learn more about this past. 
\item<3-> Thanks for your attention and I'd be happy to take questions! 
\end{itemize}
\end{frame}



%From a historian's point of view it's also interesting to ask how understanding of Euclid's Book II evolved over time
%More recent work (see Corry, Archive for History of Exact Sciences (2013)) – as the Elements passed through the Medieval Islamic world 
%and Renaissance Europe (where our symbolic algebra was developed), the connection between geometry and algebra was realized and alternate %algebraic proofs started to be presented for the propositions in Book II (!)
%“Geometric algebra” probably does do a good job of describing later understanding!
\end{document}

