\documentclass{article}
\usepackage{psfig}
\usepackage{lgreek}
\renewcommand{\baselinestretch}{1.3}
\newcommand{\itemspace}{\hspace{8pt}}
\newcommand{\hfhf}{\left(\frac{1}{2},\frac{1}{2}\right)}
\newcommand{\overp}{\overline{P}}
\def\thefigure{\arabic{section}.\arabic{subsection}.\arabic{figure}[p]}
\def\mod{{\rm mod}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Welcome to the macros of Ilan Vardi!
%%%
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%
%%%%%% 1 in margins produced by this
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{8.9in}
\setlength{\topmargin}{0pt}
\setlength{\oddsidemargin}{0pt}
\setlength{\evensidemargin}{0pt}
\setlength{\headheight}{0pt}
\setlength{\headsep}{0pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% TeX's eqalign macros....
%%%
\catcode`@=11
\newskip\@centering \@centering=0pt plus 1000pt minus 1000pt
\def\openup{\afterassignment\@penup\dimen@=}
\def\@penup{\advance\lineskip\dimen@
\advance\baselineskip\dimen@
\advance\lineskiplimit\dimen@}
\def\eqalign#1{\null\,\vcenter{\openup\jot\m@th
\ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
\crcr#1\crcr}}\,}
\newif\ifdt@p
\def\displ@y{\global\dt@ptrue\openup\jot\m@th
\everycr{\noalign{\ifdt@p \global\dt@pfalse
\vskip-\lineskiplimit \vskip\normallineskiplimit
\else \penalty\interdisplaylinepenalty \fi}}}
\def\@lign{\tabskip\z@skip\everycr{}} % restore inside \displ@y
\def\displaylines#1{\displ@y
\halign{\hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}\crcr
#1\crcr}}
\def\eqalignno#1{\displ@y \tabskip\@centering
\halign to\displaywidth{\hfil$\@lign\displaystyle{##}$\tabskip\z@skip
&$\@lign\displaystyle{{}##}$\hfil\tabskip\@centering
&\llap{$\@lign##$}\tabskip\z@skip\crcr
#1\crcr}}
\def\leqalignno#1{\displ@y \tabskip\@centering
\halign to\displaywidth{\hfil$\@lign\displaystyle{##}$\tabskip\z@skip
&$\@lign\displaystyle{{}##}$\hfil\tabskip\@centering
&\kern-\displaywidth\rlap{$\@lign##$}\tabskip\displaywidth\crcr
#1\crcr}}
\catcode`@=12
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% boxed text
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newlength{\boxedparwidth}
\setlength{\boxedparwidth}{.92 \textwidth}
\newenvironment{boxedtext}
{\begin{center}
\begin{tabular}{|@{\hspace{.15 in}}c@{\hspace{.15 in}}|}
\hline \\ \begin{minipage}[t]{\boxedparwidth}
\setlength{\parindent}{.25 in}}
{\end{minipage} \\ \\ \hline \end{tabular} \end{center}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\calc{{\cal C}}
\def\e{\varepsilon}
\begin{document}
\title{What is Ancient Mathematics?}
\author{Ilan Vardi}
\date{}
\maketitle
``In my opinion, it is not only the serious accomplishments of great and
good men which are worthy of being recorded, but also their
amusements.''
\smallskip
\begin{flushright}
Xenophon, {\sl Symposium}
\end{flushright}
\bigskip\smallskip
The title of this paper is a result of comments on earlier drafts by
mathematicians: ``This is not mathematics, this is history!'' and by
historians of mathematics: ``This is not history, this is
mathematics!'' After some reflection, I came to the conclusion that
the historians were right and the mathematicians were wrong--for
example, I have found little difference between reading papers of
Atle~Selberg (1917--, Fields~Medal~1950) and Archimedes
(287--212~B.C.) (who both lived in Syracuse!). I believe that the
mathematicians I spoke to were expressing a generally held belief that
reading mathematical papers that are over a hundred years old is
history of mathematics, not mathematics. Thus, the reconstruction of
Heegner's solution to the class number one problem (1952) appeared in
a mathematics journal \cite{Stark1} while a reconstruction of the
missing portions of Archimedes's {\sl The Method\/} (250~B.C.) appeared
in a history journal \cite{Hayashi}.
To me, reading and proving results about a mathematical
paper, whether it was written in 1950~A.D. or 250~B.C., is always
mathematics, though the latter case might be called ``ancient mathematics.''
At least as to Greece, this is accepted by some \cite[p.~21]{Hardy}:
\begin{quotation}
Oriental mathematics may be an interesting curiosity, but Greek
mathematics is the real thing\ldots
The Greeks, as Littlewood said
to me once, are not clever schoolboys or ``scholarship candidates,''
but ``Fellows of another college.'' So Greek mathematics is ``permanent,''
more perhaps even than Greek literature. Archimedes will be remembered
when Aeschylus is forgotten, because languages die and mathematical
ideas do not.
\end{quotation}
\noindent
I am saying that ancient Greek mathematicians were
in every essential way similar to modern mathematicians. In fact,
some mathematicians might find more in common with Archimedes and
Euclid than with many colleagues of their departments, and even
reading the original Greek, a subject traditionally
taught in High School \cite{Athenaze}, seems easier than understanding
say, the proof that every semistable elliptic curve is modular
\cite{Wiles}.
Nineteenth century mathematicians dedicated much of their research to
elementary Euclidean geometry.
It is possible that some mathematicians of that
era felt that the influence of the past was too great, as
Felix Klein wrote \cite[Vol.~2, p.~189]{Klein}:
\begin{quotation}
Although the Greeks worked fruitfully, not only in geometry but
also in the most varied fields of mathematics, nevertheless we
today have gone beyond them everywhere and certainly also in
geometry.
\end{quotation}
\noindent
For whatever reason,
geometers recently tend to
distance themselves from Euclidean geometry.
For example,
the book {\sl Unsolved Problems in Geometry\/} \cite{Unsolved}, part
of a series on ``unsolved problems in intuitive mathematics'' does not
have a section devoted to classical Euclidean geometry, and with few
exceptions such as \cite{Avron}, articles on this subject are
relegated to ``low--brow'' publications. Yet
earlier in this century, Bieberbach, Hadamard, and Lebesgue all wrote
books on elementary Euclidean geometry \cite{Bieberbach}
\cite{Hadamard} \cite{Lebesgue} and excellent
books and articles on
ancient mathematics are still being written \cite{Hartshorne}
\cite{Toussaint}. See \cite{Davis} for further analysis of these
issues.
In this paper, I will give an example of ancient mathematics by using
techniques that Archimedes developed in his paper
{\sl The Method\/}
to derive results that he proved in his paper {\sl On Spirals.} I
will try to present these in a way that Archimedes might understand
\cite{Vardi2}, in particular, the diagrams are intended to conform to
ancient Greek standards \cite{Netz}. I will also indicate how ideas in
these papers can lead to some surprising results (e.g., Exercise~4
below). The paper will include such exercises as may challenge the
reader to understand concepts of Archimedes as he expressed them.
I have concentrated on the works of Archimedes because these are most
similar to modern mathematical research papers, sharply
focused on problems and their solution. By comparison, the
works of Euclid read like a generic textbook; and so little is known
about Euclid that it cannot be ruled out that he was actually a
``consortium.'' Moreover, it seems likely that the works of Euclid are
based on the efforts of earlier mathematicians \cite{Fowler}
\cite{Knorr0}.\footnote{Hence I question the curriculum of
St.~John's College which purports to educate its students
by following an historical sequence of original sources. Its reading
list also includes the ancient textbook \cite{Nicomachus}.}
The balance of the paper shows how a precise knowledge of ancient
mathematics allows one to navigate in the sea of inaccuracies and
misconceptions written about the history of mathematics.
This also gives one perspective on cultural aspects of mathematics as
it forces one to understand ideas of first rate mathematicians whose
cultural background is very different from the present one. For
example, it can help you read {\sl The New York Times\/}
\cite{NYT}:
\begin{quotation}
``Alien intelligences may be so far advanced that their math
would simply be too hard for us to grasp,'' [Paul] Davies said.
``The calculus would have baffled Pythagoras, but with suitable
tuition he would have accepted it.''
\end{quotation}
\noindent
Reading this paper should make it clear that Archimedes could have
been Pythagoras's calculus tutor, thus refuting any notion that
calculus was an unknown concept to ancient Greeks.
\medskip
It is my hope that I can convince mathematicians that there are
many interesting and relevant ideas to be uncovered in
ancient Greek mathematics, and that it might be worthwhile to take a
first hand look, being wary of popular accounts and secondary sources,
this one included!
\subsection*{Extending Archimedes's Method}
In 1906 the Danish philologist J.L.~Heiberg went to Constantinople to
examine a manuscript containing mathematical writing which had been
discovered seven years earlier in the monastery of the Holy Sepulchre
at Jerusalem. What he found was a 10th--century palimpsest, a
parchment containing works of Archimedes that, sometime
between the 12th and 14th centuries, had been partially erased and
overwritten by religious text. Heiberg managed to decipher the
manuscript \cite{Heiberg1} and found that it included a text of
{\sl The Method,}
a work of Archimedes previously thought lost. (The story
of the transmission of Archimedean manuscripts given in
\cite{Dijksterhuis} reads like a chapter from {\sl The Maltese
Falcon\/}). Late bulletin: Heiberg's palimpsest was
sold in auction to an unknown buyer
for \$ 2,000,000 on October 29, 1998 \cite{Rickey2}.
This gave scholars the opportunity to examine it for the first
time in almost 100 years \cite{Rickey}.
Heiberg's discovery had a significant impact on the understanding
of ancient Greek mathematics, for two reasons. The first is
the aim of the paper, summarized by
Archimedes\footnote{Archimedes is addressing
Eratosthenes of Cyrene (circa 284--194~B.C.),
director of the library of Alexandria, famous for
his accurate measurement of the circumference of the earth
\cite{Cleomedes} and
his sieve to compute prime numbers \cite{Nicomachus}.
} \cite[Vol.~2, p.~221]{Thomas}
\begin{quotation}
Moreover, seeing in you, as I say, a zealous student and a man of
considerable eminence in philosophy, who gives due honour to
mathematical inquiries when they arise, I have thought fit
to write out for you and explain in detail in the same book
the peculiarity of a certain method, with which furnished
you will be able to make a beginning in the investigation
by mechanics of some of the problems in mathematics. I am
persuaded that this method is no less useful even for the proofs
of the theorems themselves. For some things first became clear
to me by mechanics, though they had later to be proved geometrically
owing to the fact that investigation by this method does not
amount to actual proof; but it is, of course, easier to provide
the proof when some knowledge of the things sought has been
acquired by this method rather than to seek it with no prior
knowledge.
\end{quotation}
\noindent
This is a radical divergence from all other extant Greek works,
as T.L.~Heath explains \cite[Supplement, p.~6]{Heath}:
\begin{quotation}
Nothing is more characteristic of the classical works of the
great geometers of Greece, or more tantalising, than the
absence of any indication of the steps by which they worked
their ways to the discovery of their great theorems. As they
have come down to us, these theorems are finished masterpieces
which leave no traces of any rough--hewn stage, no hint of
the method by which they were evolved\ldots A partial
exception is now furnished by the {\sl Method}; for here
we have a sort of lifting of the veil, a glimpse of the
interior of Archimedes's workshop as it were.
\end{quotation}
\noindent
The other surprising aspect of {\sl The Method\/} is the
revelation that Archimedes worked with {\sl infinitesimals,} for example,
``The triangle $\Gamma ZA$
is composed of the straight lines drawn in $\Gamma ZA$,''
\cite[Vol.~2, p.~227]{Thomas},
``The cylinder, the sphere
and the cone being filled by circles thus taken\ldots''
\cite[Vol.~3, p.~91]{Mugler}, see \cite{AB} \cite{Knorr3}.
As every mathematician knows,
infinitesimals were reinvented by mathematicians such as
Cavalieri (1598--1647) and Leibniz (1646--1716), see \cite{Andersen}
\cite{Edwards}.
Archimedes used them to compute the area and volumes of various
geometrical figures including what he considered his greatest
achievement:\footnote{Archimedes requested that a diagram of a sphere
inscribed in a cylinder along with their proportion be placed on his grave,
which Cicero reported finding in 75~B.C. when he was treasurer of
Sicily \cite[Vol.~2, p.~33]{Thomas}.}
\smallskip\noindent
{\sl any cylinder having for its base the greatest of the circles
in the sphere, and having its height equal to the diameter
of the sphere, is one--and--a--half times the sphere,
}
\smallskip\noindent
a result he subsequently proved,
{\sl On the Sphere and Cylinder,~I,}
Corollary to Proposition~34
\cite[Vol.~2, p.~125]{Thomas}.
Archimedes understood that his method does not produce valid proofs
due to its use of
infinitesimals,\footnote{In {\sl The Quadrature of the Parabola\/}
Archimedes gave what he considered to be a rigorous proof using
the mechanical method
of a result conjectured in a similar way in {\sl The Method,} but
using infinitesimals.}
though it is unclear if the same is true
of his successors. In any case, it is easy to make the arguments rigorous,
given present knowledge. The basic ideas of {\sl The Method\/} are
still presented in contemporary calculus courses \cite{Harvard}
\cite[p.~709]{Simmons}, and
a physical model of Archimedes's argument has been built \cite{Gould}.
\medskip
On the other hand, Archimedes's
\underline{\sl On Spirals\/} is a masterpiece of rigorous
mathematics. In this paper, Archimedes computes the area and tangent
of a spiral, and, in doing so, derives much of the Calculus~I
curriculum, including related rates, limits, tangents, and the
evaluation of Riemann sums. This is reflected by the fact that a
number of contemporary Calculus texts outline the basic idea of
Archimedes's computation of the area of a spiral \cite[p.~3]{Apostol}
\cite[p.~75]{Bers}, though both these works avoid technical
difficulties by substituting a parabola, but then incorrectly imply
that Archimedes used such an approach for the parabola
\cite[p.~8]{Apostol} \cite[p.~75]{Bers}.
The considerable length of the paper is a consequence of proving
these results from basic principles. Unfortunately, it does not yet
have a faithful English translation \cite{Vardi}; Heath's intent
in \cite{Heath} was to capture the modern flavor of Archimedes's works
in order to make them more accessible. A generally
faithful French translation,
including the Greek text, is available \cite{Mugler}.
%Moreover, Heath's translation is based on an earlier version
%of Heiberg updated edition \cite{Heiberg} on which all
%subsequent translations have been based, e.g.,
%the French translation \cite{Mugler} which also reproduces Heiberg's
%Greek edition.\footnote{One might wonder why Heiberg's edition
%contains a Latin translation since this is also a
%dead language. This follows a tradition based on the later transmission of
%Greek works into Europe, roughly corresponding to the fall of
%Constantinople in 1453.}
\medskip
The mechanical method does not seem to produce directly the area of a
spiral, or even the area of a circle also computed by Archimedes so
one might wonder how he first derived them. W.R.~Knorr \cite{Knorr1}
has suggested that the writings of Pappus of Alexandria (fourth Century
AD) indicate that Archimedes wrote an earlier version of
{\sl On Spirals\/}
which used a different argument to compute the area of the
spiral but then rejected it as inelegant
(this approach is developed in the solution to Exercise~1).
The object of the next section is to show how a natural extension
of the mechanical method easily produces these results.
%It is therefore my answer to the question:
%``If you could meet anyone who ever lived, who would it be and
% what would you say to them?''\footnote{I conjecture that Archimedes's
%response might be: ``Move away from my diagram.''}
%see \cite{Moise} for a completely
%rigorous development of these, and \cite{Netz} for an analysis
%of his diagrams.
\subsection*{Weighing a spiral}
{\sl The Method\/} relies on a mechanical analogy by using a balance
to compare objects. This requires a few simple assumptions and
facts about the properties of a lever, which are developed
(sometimes implicitly) in Archimedes's
{\sl On the Equilibrium of Planes~I}, \cite{Heath}
\cite[Chapter~IX]{Dijksterhuis}. These can be summarized by
\begin{description}
\item[{\bf Assumption 1.}] Two objects will balance
each other if the distances of their center of gravity to the fulcrum
are inversely proportional to their weight. The center of gravity of
an object lies on an axis of symmetry.
\smallskip\noindent When only the weight of an object is relevant to
an argument, I will place it on a {\sl pan\/} suspended from the
balance. The object and any of its sections will then be assumed to have
their center of gravity at the point where the pan is suspended. I
will also make extra assumptions not seen in Archimedes's works
(however, see the solution to Exercise~4)
\medskip\noindent
\item[{\bf Assumption 2.}] A plane figure is composed of
circular arcs with common center and each circular arc weighs the same
as a line segment of equal length.
\end{description}
\medskip
{\bf Exercise 1.} What happens if you instead
decompose plane figures into radii with common center?
\smallskip\noindent
I will first show how the mechanical method can be used to derive
Archimedes's formula for the area of a circle given in
{\sl Measurement
of the Circle,} Proposition~1 \cite[Vol.~1, p.~317]{Thomas}
(a similar method was used by Rabbi Abraham bar Hiyya (1070--1136),
see \cite{Katz} \cite{Sanchez}).
\medskip\noindent
{\bf Proposition 1.}
{\sl Any circle is equal to a right--angled triangle in which
one of the sides about the right angle is equal to the radius,
and the base is equal to the circumference.}
\smallskip
{\bf Exercise 2.} Explain why Proposition 1 is equivalent to the
familiar formula: Area of a circle=$\pi R^2$.
\vspace{.3in}
\begin{center}
{\bf Figure 1.}
\end{center}
\vspace{.3in}
\medskip\noindent Suspend two pans on opposite sides of a balance
and at equal distances to the fulcrum. On one pan,
place a circle with center at $A$ and radius $AB$, on the other
place a line segment $CD$ of length $AB$.
By Assumption~2, the
circle is composed of circumferences with center $A$ and radius $AE$ for any
$E$ lying on $AB$. For each such circumference, place a
line segment $FG$ perpendicular to $CD$, of length the circumference
through $E$ such that its endpoint $F$ lies on $CD$ and $CF$ is equal to
$AE$. By Assumption~2, the line segment $FG$ is in equilibrium with
the circumference through $E$. The resulting figure is a right triangle of
height $AB$, base the circumference through $B$, and it balances
a circle of radius $AB$, which is the statement of Proposition~1.
\smallskip
{\bf Exercise 3.} Why is the resulting figure in this construction
a triangle?
\smallskip
{\bf Exercise 4.} Generalize the following
heuristic from {\sl The Method\/} \cite[Supplement]{Heath}:
``\ldots judging from the fact that any circle is equal to a triangle
with base equal to the circumference and height equal to the
radius of the circle, I apprehended that, in like manner, any
sphere is equal to a cone with base equal to the surface of the
sphere and height equal to the radius.''
\vspace{.3in}
\begin{center}
{\bf Figure 2.}
\end{center}
\vspace{.3in}
\medskip\noindent
Archimedes's definition of a spiral and its relevant components
is given by \cite[Vol.~2, p.~183]{Thomas}:
\begin{enumerate}
\item
If a straight line drawn in a plane revolve uniformly
any number of times about a fixed extremity until it return
to its original position, and if, at the same time as the
line revolves, a point move uniformly along the straight line,
beginning at the fixed extremity, the point will describe a
{\sl spiral\/} in the plane.
\item
Let the extremity of the straight line which remains fixed while the
straight line revolves be called the {\sl origin\/} of the spiral.
\item
Let the position of the line, from which the straight line began
to revolve, be called the {\sl initial line\/} of the revolution.
\end{enumerate}
\vspace{.3in}
\begin{center}
{\bf Figure 3.}
\end{center}
\vspace{.3in}
\medskip\noindent {\bf Proposition 2.} {\sl The area inside a
spiral anywhere within its first revolution is one third the sector of
a circle with center at the origin of the spiral,
radius equal to the distance
of the point describing the spiral to the origin, and angle equal to
the angle between the line and the initial line.
} (Archimedes gave areas for complete revolutions only, but
his proof also applies to this case.)
\medskip\noindent Consider a spiral with origin $A$,
initial line $AB$, and $C$ the position of the point describing
the spiral. Consider also a balance arm $DE$ of length twice $AC$ and let
the midpoint $F$ of $DE$ be the fulcrum.
On this balance suspend a pan from
$D$ and place the spiral region in the pan.
\vspace{.3in}
\begin{center}
{\bf Figure 4.}
\end{center}
\vspace{.3in}
By Assumption~2, the spiral region is composed of arcs $GH$ for each
$G$ lying on $AC$, where $H$ is the intersection of the circle with
center $A$ and radius $AG$ and the spiral. Extend $AH$ to intersect
the circle of center $A$ and radius $AC$ at $I$. Consider
a line segment $JK$ of length equal to the
arc $CI$ and crossing $DE$ at $L$ such that $JK$ and $DE$ are
perpendicular, $L$ is the midpoint of $JK$, and $FL$ is equal to $AG$.
I claim that $JK$ and the arc $GH$ are in equilibrium.
To see this
note that, by Exercise~3, the length of an arc is proportional to
its radius so that
$$
\mbox{arc $GH$} : \mbox{arc $CI$} :: AG : AC,
$$
and the result follows from the assumption that the
arc $CI$ has its center of gravity at $D$ and from Assumption~1.
Now extend the arc $CI$ to intersect $AB$ at $M$; then
the arc $CI$ is equal to the arc $CIM$ minus the arc $IM$, and by
the definition of spiral, $IM$ is proportional to $AH$. Since
the arc $CIM$ remains constant in this argument, the second part of
Exercise~3 shows that the arc~$CIM$ minus $JK$ is proportional to $FL$,
which means that the resulting figure is an isosceles triangle
which balances the inside of the spiral.
\vspace{.3in}
\begin{center}
{\bf Figure 5.}
\end{center}
\vspace{.3in}
The exact same argument shows that the area between the spiral and the
initial line that lies within the same sector balances
the same isosceles triangle, but reversed so that its vertex lies on
the fulcrum. The crucial step is to recall the following
\medskip {\bf Fact:} The center of gravity of a triangle
lies at the intersection of the medians, and the medians of a triangle
intersect each other in a ratio of $2:1$.
\medskip\noindent The first part is suggested by the observation
that a median divides a triangle into two triangles of equal weight
and its proof is one of
the main results of {\sl On the Equilibrium of Planes~I.} The second
part is an easy exercise \cite[\S 1.4]{Coxeter} and
follows from {\sl On the Equilibrium of Planes~I,}
Proposition~15, generalized to trapezoids.
\medskip\noindent
This shows that reversing the first triangle places the
center of gravity twice as far from the fulcrum so the second triangle
will balance twice the first.
One concludes that the inside of the spiral weighs one half the outside
of the spiral and thus one third of the sector of the circle, which is
the statement of Proposition~2.
\smallskip
{\bf Exercise 5.} Evaluate the area of the spiral
using the procedure as for the circle, i.e., by only comparing weights
placed on pans.
\smallskip {\bf Exercise 6.} Use the mechanical method
to compute the center of gravity of a spiral region.
\subsection*{A modern translation}
\vspace{.3in}
\begin{center}
{\bf Figure 6.}
\end{center}
\vspace{.3in}
The basic observation is that Assumption~2 extends Archimedes's
method to polar coordinates. Consider a curve $r = f(\theta)$,
in polar coordinates where, for simplicity, $f(r)$ is an increasing
function, so there is an inverse function $\theta = g(r)$
(this notation is more convenient given the difficulties of Exercise~1).
To compute the area of a region ${\cal A}$ lying
inside the curve and having
$0\le \theta \le \Theta$, one partitions ${\cal A}$ into
thin circular shells of width $h>0$, as in Figure~6.
Using the formula $\theta r^2/2$ for the area of a sector of angle
$\theta$ and radius $r$, each shell has area
$
(\Theta - \theta) r h + R(r, h)\,,
$
where the error $R(r, h)$ is less than the area of the small shell
element of area
$(r + h)h[g(r+h) - g(r)]$, see Figure~6, and this is less than
$C r h^2$, for some constant $C$,
assuming that $g(r)$ is well behaved. It follows that, ignoring
terms of order $h^2$,
the area of each shell is $(\Theta - \theta) rh$, which
is the length of the bottom arc of the shell multiplied by $h$. This
shows why the first part of Assumption~2 holds. All these shells have
area a linear function of $h$ up to an error term of lower order, and
form a disjoint union of ${\cal A}$, which shows why the second part of
Assumption~2 holds. Letting $h\to 0$, it follows that the area
of ${\cal A}$ is
$$
\int_{0}^R (\Theta - \theta) r dr
=
\int_{0}^R [g(R) - g(r)] r dr\,.
$$
The standard derivation of this formula uses the formula
$rdrd\theta$ for the area element in polar coordinates
$$
\mbox{Area of }{\cal A} =
\mathop{\int\!\!\int}_{\!\cal A\;\;} dxdy
=
\mathop{\int\!\!\int}_{\!\cal A\;\;} rdrd\theta
=
\int_{r=0}^R \int_{\theta = g(r)}^\Theta r d\theta dr
=
\int_{0}^R [g(R) - g(r)]\, r dr\,.
$$
A circle is simply $g(r) = 0$ which yields
$
2\pi \int_{0}^R r dr = \pi R^2\,.
$
\bigskip
A spiral, in polar coordinates, is given by the equation $r = a\theta$,
for some constant $a$, so can be written as
$\theta = k r$, where $k = 1/a$.
By the above, the area of the spiral is
$$
\int_0^R (kR - kr) r dr
=
kR \, \int_0^R r dr - k\, \int_0^R r^2 dr
=
\frac{k R^3}{2} - k \, \frac{R^3}{3}
=
\frac{kR^3}{6}
=
\frac{1}{3}\, \frac{\Theta R^2}{2}\,,
$$
where the term on the right is seen to be $1/3$ the area of the sector
of the circle of radius $R$ and angle $\Theta$, yielding Proposition~2.
Any proof of this formula is equivalent to evaluating such integrals.
Archimedes evaluated $\int_0^R r^2 dr$ by decomposing it into Riemann
sums and obtaining a closed form for the sum $1^2 +\cdots + n^2$. In
Section~2 this integral is computed by realizing it as the moment of a
triangle and evaluating this as its weight multiplied by the distance
of its center of gravity from the fulcrum.
\subsection*{The Way of Archimedes}
The Calculus Reform movement has emphasized experimentation over rigor
in calculus education and has been criticized as a result
\cite{Swann}. To defend its position that physical problems should be
used to discover mathematical results Harvard Calculus appeals to
Archimedes and {\sl The Method\/} \cite[p.~vii]{Harvard}
\smallskip
{\bf The Way of Archimedes:} {\sl Formal definitions and procedures
evolve from the investigation of practical problems.}
\medskip\noindent
This principle accurately represents the works of Archimedes, but a
disparity arises in that Harvard Calculus postpones
mathematical rigor indefinitely, so Archimedes's
name should be among the last to be associated with such an
endeavor. For example, the method of exhaustion used by Archimedes is
essentially the $\e$--$\delta$ argument abandoned by Harvard Calculus,
as B.L.~van~der~Waerden writes \cite[p.~220]{Waerden}:
\begin{quotation}
\ldots the estimations, which occur in the summing of infinite
series and in limit operations, the `epsilontics', as the calculation
with an arbitrary small $\e$ is sometimes called, were for Archimedes
an open book. In this respect, his thinking is entirely modern.
\end{quotation}
\smallskip\noindent
Moreover, Archimedes held in contempt those who did not furnish proofs
of their results. In the introduction to {\sl On Spirals,} Archimedes
reveals that he intentionally announced false theorems in order to
expose some of his contemporaries \cite{Heath}
\begin{quotation}
\ldots I wish now to put them in review one by one, particularly as it
happens that there are two among them which [are wrong and which
may serve as a warning to] those who claim to discover everything but
produce no proofs of the same may be confuted as having actually
pretended to discover the impossible.
\end{quotation}
\medskip\noindent Harvard Calculus fails miserably when measured
against this Way of Archimedes. Apart from the passage quoted
above, the word ``theorem'' appears in \cite{Harvard} only in the name
``Fundamental Theorem of Calculus.'' Compare this with a standard
calculus text \cite{EP} which lists 130 theorems in its index. Even
more revealing, the only instance of the word ``proof'' I
located in \cite{Harvard} was in Archimedes's introduction to the method
quoted above and used in \cite{Harvard} to justify ``The Way of Archimedes.''
In fact, this quote emphasizes that discovery of the answer to a
problem leads to a {\sl theorem\/} whose {\sl proof\/} is facilitated
by knowledge of the answer. My interpretation is not Calculus Reform
but
\smallskip {\bf Problem Solving:} When faced with a problem,
use {\sl any\/} method that allows you to conjecture the answer, then
find a rigorous proof.
\medskip\noindent
{\bf A recent development:} The second edition of \cite{Harvard}
has taken a more moderate approach to Calculus Reform and
now includes some complete proofs \cite[2nd Edition, p.~78]{Harvard}
and the $\e$--$\delta$ definition of a limit
\cite[2nd Edition, p.~128]{Harvard}.
However, this new edition no longer includes ``The Way of Archimedes.''
\subsection*{Popular Misconceptions}
It must be noted that the
penultimate remark of the previous section
paraphrases E.T.~Bell \cite[p.~31]{Bell}: ``In short he used
mechanics to advance his mathematics. This is one of his titles to a
modern mind: {\sl he used anything and everything that suggested
itself as a weapon to attack his problems.''} However, strong
opinions such as those expressed in \cite{Bell} are fraught with
danger, and it is instructive to include the continuation of this
passage
\begin{quotation}
To a modern all is fair in war, love, and mathematics; to many of
the ancients, mathematics was a stultified game to be played according
to the prim rules imposed by the philosophically--minded Plato. According
to Plato only a straightedge and a pair of compasses were to be permitted
as the implements of construction in geometry. No wonder the classical
geometers hammered their heads for centuries against 'the three
problems of antiquity': to trisect an angle; to construct a cube
having double the volume of a given cube; to construct a square equal
to a circle.
\end{quotation}
%\noindent
%Once again this relies on Plutarch account of Archimedes
%\cite{Plutarch}
%(the problem of mean proportions is equivalent to doubling the
%cube, see \cite[p.~3]{Knorr2} \cite[Vol.~1, p.~258]{Thomas})
%
%\begin{quotation}
%Eudoxus and Archytas had been the first originators of this far--famed
%and highly--prized art of mechanics, which they employed as an elegant
%illustration of geometrical truths, and as means of sustaining
%experimentally, to the satisfaction of the senses, conclusions too
%intricate for proof by words and diagrams. As, for example, to solve
%the problem, so often required in constructing geometrical figures,
%given the two extremes, to find the two mean lines of a proportion,
%both these mathematicians had recourse to the aid of instruments,
%adapting to their purposes certain curves and sections of lines. But
%what with Plato's indignation at it, and his invectives against it as
%the mere corruption and annihilation of the one good of geometry,
%which was thus shamefully turning its back upon the unembodied objects
%of pure intelligence to recur to sensation, and to ask help (not to be
%obtained without base supervisions and depravation) from matter; so it
%was that mechanics came to be separated from geometry, and, repudiated
%and neglected by philosophers, took its place as a military art.
%\end{quotation}
\noindent
This has since been discredited, see
\cite{Fowler} \cite{Knorr2} (better yet, look at original sources,
e.g., as collected in \cite[Vol.~1, Chapter~9]{Thomas}), and
van~der~Waerden writes \cite[p.~263]{Waerden},
\begin{quotation}
The idea, sometimes expressed, that the Greeks only permitted
constructions by means of compasses and straight edge, is
inadmissible. It is contradicted by the numerous constructions,
which have been handed down, for the duplication of the cube and
the trisection of the angle.
\end{quotation}
\noindent
In particular, Archimedes trisected the angle with ruler
and compass in Proposition~8 of {\sl The Book of Lemmas\/}
\cite[p.~309]{Heath}, see \cite{Dudley} \cite[Section~31]{Hartshorne}.
The history of this misconception might prove an
interesting subject for further study.
Unfortunately, it is only one of a number of popular misconceptions
about the limitations of Greek science \cite{Vardi}.
For example, Isaac~Asimov (1920--1992) has written \cite{Asimov}
\begin{quotation}
To the Greeks, experimentation seemed irrelevant. It interfered with
and detracted from the beauty of pure deduction\ldots To test a
perfect theory with imperfect instruments did not impress the Greek
philosophers as a valid way to gain knowledge\ldots
The Greek rationalization for the ``cult of uselessness'' may
similarly have been based on a feeling that to allow mundane
knowledge (such as the distance from Athens to Corinth) to intrude
on abstract thought was to allow imperfection to enter the Eden
of true philosophy.
Whatever the rationalization, the Greek thinkers
were severely limited by their attitude.
Greece was not barren of
practical contributions to civilization, but even its great engineer,
Archimedes of Syracuse, refused to write about his inventions and
discoveries\ldots
to maintain his amateur status, he broadcast only his
achievements in pure mathematics.
\end{quotation}
\noindent
This passage is contradicted by numerous examples of Greek
scientific experiments, for example, Eratosthenes's measurement
of the earth \cite{Asimov3}. Asimov may be excused for
paraphrasing Plutarch's account of Archimedes in his
{\sl Life of Marcellus,} written circa 75~AD \cite{Plutarch}
\cite[Vol.~2, p.~31]{Thomas}
\begin{quotation}
Yet Archimedes possessed so lofty a spirit, so profound a soul,
and such a wealth of scientific inquiry, that although he had acquired
through his inventions a name and reputation for divine rather than
human intelligence, he would not deign to leave behind a single writing
on such subjects. Regarding the business of mechanics and every utilitarian
art as ignoble or vulgar, he gave his zealous devotion only to those
subjects who elegance and subtlety are untrammeled by the necessities
of life\ldots
\end{quotation}
%\begin{quotation}
%Yet Archimedes possessed so high a spirit, so profound a soul, and such
%treasures of scientific knowledge, that though these inventions
%had now obtained him the renown of more than human sagacity, he
%yet would not deign to leave behind him any commentary or writing
%on such subjects; but, repudiating as sordid and
%ignoble the whole trade of engineering, and every sort of art that lends
%itself to mere use and profit, he placed his whole affection and
%ambition in those purer speculations where there can be no reference
%to the vulgar needs of life\ldots
%\end{quotation}
\noindent
Despite Plutarch's ancient credentials
he had no better insight into Archimedes's
scientific contributions which contradict his story.
The reader is already aware that {\sl The Method\/} shows that
physical considerations played an important role in Greek mathematics.
But Asimov and Plutarch are completely refuted by
Archimedes in {\sl The Sand
Reckoner\/} \cite{Heath} \cite{Dijksterhuis}
\begin{quotation}
While examining this question I have, for my part tried in the
following manner, to show with the aid of instruments, the angle
subtended by the sun, having its vertex at the eye. Clearly, the exact
evaluation of this angle is not easy since neither vision, hands, nor
the instruments required to measure this angle are reliable enough to
measure it precisely. But this does not seem to me to be the place to
discuss this question at length, especially because observations of
this type have often been reported.
For the purposes of my proposition, it suffices to find an
angle that is not greater than the angle subtended at the sun with
vertex at the eye and to then find another angle which is not less
than the angle subtended by the sun with vertex at the eye.
A long ruler having been placed on a vertical stand placed in the
direction the rising sun is seen, a little cylinder
was put vertically on the ruler immediately after sunrise. The
sun, being at the horizon, can be looked at directly, and the ruler
is oriented towards the sun and the eye placed at the end of the
ruler. The cylinder being placed between the sun and the eye, occludes
the sun. The cylinder is then moved further away from the eye and as
soon as a small piece of the sun begins to show itself from each side
of the cylinder, it is fixed.
If the eye were really to see from one point, tangents to the cylinder
produced from the end of the ruler where the eye was placed would make
an angle less than the angle subtended by the sun with vertex at the
eye. But since the eyes do not see from a unique point, but from a
certain size, one takes a certain size, of round shape, not smaller
than the eye and one places it at the extremity of the ruler where the
eye was placed\ldots
the width of cylinders producing this effect is not smaller
than the dimensions of the eye.
\ldots It is therefore clear that
the angle subtended by the sun with vertex at the eye is also
smaller than the one hundred and sixty fourth part of a right angle,
and greater than the two hundredth part of a right angle.
\end{quotation}
\noindent
The correct value of the angular diameter of the sun is now known to
average about $34'$ \cite[p.~95]{Green}, i.e., the 159th part of a
right angle. It is important to note that this shows not only that
ancient Greeks frequently performed experiments, but that Archimedes
dealt with experimental error and also compensated for the fact that
the human eye is part of the observational instrument, thus
anticipating scientists such as Hermann von Helmholtz
(1821--1894) \cite{Helmholtz}. A translation and analysis of {\sl The
Sand Reckoner\/} is given in \cite{Vardi}.
\subsection*{Answers to Exercises}
\medskip\noindent {\bf Exercise 1.} A naive approach leads to
incorrect results, evidence of the dangers of using infinitesimals, and
indicating why Archimedes did not consider his method to be
rigorous. For example, taking the radii of a circle of radius $R$,
with respect to the circumference, and reordering them to form a
rectangle yields area $2\pi R^2$. For a general figure, it's not even
clear how to pick the radii. To make sense of what is going on, one
regards radii as limits
of sectors, i.e., infinitesimal triangles. In the case of the circle,
this means that the weight of a radius, with respect to the
circumference, is equal to {\sl one half\/} its length. This can be
loosely interpreted as the argument Archimedes used to compute the
area of the circle \cite{AB}. In the general case, the
following is justified:
\smallskip
{\bf Assumption 3.} The weight of a radius is proportional to
the square of its length.
\medskip\noindent
In modern notation, this is simply
$$
\mathop{\int\!\!\int}_{\!\cal A\;\;} rdrd\theta
=
\int_{\theta=0}^\Theta \int_{r = 0}^{f(\theta)} r dr d\theta
=
\frac{1}{2}\, \int_{0}^\Theta [f(\theta)]^2 d\theta\,,
$$
where the radii have been chosen with respect to the unit circle.
Given Assumption~3, one can compute the area of the spiral by using
Pappus's argument \cite[Book~4, Proposition~21]{Pappus}, see also
\cite[p.~377]{Heath2} \cite[p.~162]{Knorr2}.
\vspace{.3in}
\begin{center}
{\bf Figure 7.}
\end{center}
\vspace{.3in}
To compute the weight of a spiral region, take each radius of the
spiral, starting from the final radius, and place a disk with diameter
equal to this radius at height the current angle so the resulting
figure is a cone. Similarly, for each radius of the
sector place a disk with diameter equal to this radius at height the
current angle, resulting in a cylinder with the same base and height
as the cone.
Since Euclid's Proposition~2 of Book~12 proves that ``circles are to
one another as the squares on the diameter,'' Assumption~3 shows that
the ratio of the weight of the spiral region to the weight of the
sector is the same as the ratio of the volume of the cone to the
volume of the cylinder. But Euclid's Proposition~10 of Book~12 proves
that the volume of a cone is one third the cylinder with same base and
height, so the spiral weighs one third of the sector, which is the
statement of Proposition~2. (Note that equilateral triangles could have
been used instead of circles resulting in a pyramid whose volume is
easier to compute.)
Knorr \cite{Knorr1} comments that this appeal to three dimensional
figures might have been considered inelegant by Archimedes as it uses
volumes to compute areas. On the other hand, reversing
this argument and using the evaluation above shows that
the volume of a cone can be computed by the mechanical method,
a result which does not appear in {\sl The Method.}
\medskip\noindent {\bf Exercise 2.}
In modern notation, Archimedes's
formulation of Proposition~1 is
$
\mbox{Area of circle of radius $R$} = \displaystyle \int_0^R 2\pi rdr\,,
$
for the integral represents the area of a right triangle
with base $R$ and height $2\pi R$.
\medskip\noindent
{\bf Exercise 3.} This is equivalent to the fact that
the length of an arc of fixed angle is proportional to its radius.
In particular, $\pi$ exists, see \cite{Moise} \cite{Vardi}.
The proof is similar to \cite[Book~12, Proposition~2]{Euclid}
cited in Exercise~1, and is implicit in Archimedes's
{\sl Measurement of the Circle.} Similarly,
the length of an arc of fixed radius is proportional to its
angle.
\medskip\noindent {\bf Exercise 4.} By analogy with Assumption~2,
consider a sphere as being composed of spherical shells centered at
the center of the sphere, where each shell weighs the same as a circle
of equal area. The justification follows exactly as in Proposition~2:
Consider two pans suspended at equal distances from the fulcrum of a
balance. On one pan, place a sphere of center $A$ and radius $AB$ and
on the other a line $CD$ of length equal to $AB$. For each $E$ on $AB$
there is a spherical shell passing through $E$, and consider a circle
of area equal to this spherical shell with center at $F$ lying on
$CD$, where $CF$ equals $AE$, and such that the circle is perpendicular
to $CD$. The resulting figure is a cone with base the area of the
sphere and height the radius of the sphere; since it
balances the sphere, the claim is justified.
The similarity of this argument to the one of Proposition~1 suggests
that Archimedes may have been implicitly aware of the ideas of this
paper. Moreover, the reader may verify that the heuristic of
this exercise and its justification directly generalize to
higher dimensions
(a different generalization is given in \cite{DV}):
\medskip {\bf Proposition 3.}
{\sl The volume of an
$n$--dimensional ball is equal to the volume of a cone whose base has
$n-1$--dimensional volume equal to the $n-1$--dimensional volume of
the boundary of the ball and height equal to the radius of the ball.
}
\vspace{.3in}
\begin{center}
{\bf Figure 8.}
\end{center}
\vspace{.3in}
\medskip\noindent
{\bf Exercise 5.} The procedure when applied to the
spiral, yields a section of a parabola. The general formula for such
areas was computed by Archimedes in {\sl The Quadrature of the
Parabola,} and in this case it states that the resulting area is
four--thirds the triangle with same base and height as the section of the
parabola. Since the height and base are equal to the
final radius and half the final radius, respectively, Proposition~2
follows.
{\bf Exercise 6.}
Further extensions of Archimedes's method could be a subject for
investigation. As Archimedes wrote in {\sl The Method} \cite[Supplement,
p.~13]{Heath},
\begin{quotation}
I deem it necessary to expound the method partly because I have
already spoken of it but equally because I am persuaded that it will
be of no little service to mathematics; for I apprehend that some,
either of my contemporaries or of my successors, will, by means of
the method when once established, be able to discover other theorems
in addition, which have not yet occurred to me.
\end{quotation}
%``Let $M$ be a Riemannian [resp. Kahler or quaternionic Kahler]
%manifold such that for all $m$ in $M$, all geodesics $c$
%through $m$ and all sufficiently small $r$, the ratio between
%the volume of a geodesic sphere of radius $r$ centered at $m$ and
%the volume of a tube about $c$ of radius $r$ is the same as the
% corresponding ratio for a space form [resp. complex or quaternionic
%space form] $M_0$. Then $M$ is locally isometric to $M_0$.''
%\smallskip
%The one sided
%interpretation given by Harvard Calculus is analogous to preparing
%students for an exam by teaching test taking techniques without ever
%learning the material.
{\small
\begin{thebibliography}{xxxxxxxxx}
\bibitem{AB}
A. Aaboe and J.L. Berggren, {\sl Didactical and other remarks
on some theorems of Archimedes and infinitesimals,}
Centaurus {\bf 38} (1996), 295--316.
\bibitem{Andersen}
K. Andersen, {\sl Cavalieri's method of indivisibles,}
Arch. Hist. Exact. Sci. {\bf 31} (1985), 291--367.
\bibitem{Apostol} T. Apostol, {\sl Calculus, vol.~I, 2nd Edition,}
John Wiley \& Sons, New York, 1967.
\bibitem{Asimov3}
I. Asimov, {\sl How did we Find out that the Earth is Round?,}
Walker \& Co., New York 1972.
\bibitem{Asimov}
I. Asimov, {\sl Asimov's new Guide to Science,} Basic Books, New York 1984.
\bibitem{Heath}
Archimedes, {\sl The Works of Archimedes,} edited in modern notation
with introductory chapters by T.L. Heath. With a supplement,
{\sl The method
of Archimedes, recently discovered by Heiberg,} Dover, New York, 1953.
Reprinted (translation only) in \cite{Hutchins}.
\bibitem{Heiberg}
Archimedes, {\sl Opera Omnia, IV~vol.,} cum commentariis Eutocii, iterum
edidit I.L.~Heiberg, corrigenda adiecit E.S.~Stamatis,
B.G.~Teubner, Stuttgart, 1972.
%\bibitem{Ayoub}
%R. Ayoub, {\sl Paolo Ruffini's contributions to the quintic,}
%Arch. Hist. Exact. Sci. {\bf 23} (1980), 253--277.
\bibitem{Mugler}
Archim\`ede, {\sl Oeuvres, 4 vol.,} texte \'etabli et traduit
par C.~Mugler, Les Belles Lettres, Paris, 1970--72.
\bibitem{Athenaze}
M. Balme and G. Lawall, {\sl Athenaze, An Introduction to Ancient Greek,
2~vols.,} Oxford University Press, New York, 1990.
\bibitem{Avron}
A. Avron, {\sl On strict constructibility with a compass
alone,} J. Geometry {\bf 38} (1990), 12--15.
\bibitem{Bell}
E.T. Bell, {\sl Men of Mathematics,}
Simon and Schuster, New York, 1937.
\bibitem{Bers}
L. Bers, {\sl Calculus, vol. I,} Holt, Rinehart and Winston,
New York, 1969.
\bibitem{Bieberbach}
L. Bieberbach (1886--1982),
{\sl Theorie der geometrischen Konstruktionen,}
Lehrb\"ucher und Monographien aus dem Gebiete der exakten Wissenschaften
Mathematische Reihe, Band 13, Verlag Birkh\"auser, Basel, 1952.
\bibitem{Cleomedes}
Cleomedes (circa 150~B.C.),
{\sl De Motu Circulari Corporum Caelestium, II vols.,}
H. Ziegler, editor, Teubner, Leipzig 1891. See
\cite[p.~106]{Heath2} \cite{Lasky} \cite[Vol.~2, p.~267]{Thomas}.
\bibitem{Coxeter}
H.S.M. Coxeter, {\sl Introduction to Geometry,}
John Wiley \& Sons, New York, 1989.
\bibitem{Unsolved}
H.T. Croft, K.J. Falconer, and R.K. Guy, {\sl Unsolved Problem in Geometry,}
Springer Verlag, New York 1991.
\bibitem{Davis}
P.J. Davis, {\sl The rise, fall, and possible transfiguration
of triangle geometry: A mini history,} Amer. Math. Monthly {\bf 102}
(1995), 204--214.
%\bibitem{Davis2}
%P.J. Davis, {\sl Spirals: from Theodorus to chaos,}
%A~K~Peters, Wellesley, MA, 1993.
\bibitem{Dijksterhuis}
E.J. Dijksterhuis, {\sl Archimedes,} Princeton University Press,
Princeton 1987.
\bibitem{DV}
M. Djori\'c and L. Vanhecke,
{\sl A Theorem of Archimedes about spheres and cylinders and
two-point homogeneous spaces,}
Bull. Austral. Math. Soc. {\bf 43} (1991), 283--294.
\bibitem{Dudley}
U. Dudley, {\sl A budget of trisections,}
Springer--Verlag, New York, 1987.
\bibitem{Edwards}
C.H. Edwards, {\sl The Historical Development of the Calculus,}
Springer--Verlag, New York, 1979.
\bibitem{EP}
C.H.~Edwards and D.E.~Penney, {\sl Calculus and Analytic Geometry,
second edition,} Prentice Hall, Englewood Cliffs, NJ, 1988.
\bibitem{Euclid}
Euclid, {\sl The Thirteen Books of Euclid's Elements,} translated
with introduction and commentary by T.L. Heath, Dover 1956.
\bibitem{Fowler}
D.H. Fowler, {\sl The Mathematics of Plato's Academy: a New
Reconstruction,} Clarendon Press, Oxford 1990.
%\bibitem{Grande}
%J. Del Grande, {\sl The method of Archimedes,} Mathematics
%Teacher {\bf 86}, 240--243.
\bibitem{Gould}
S.H. Gould, {\sl The Method of Archimedes,}
Amer. Math. Monthly {\bf 62} (1955), 473--476.
\bibitem{Green}
R.M. Green, {\sl Spherical Astronomy,} Cambridge University Press,
Cambridge 1985.
\bibitem{Hadamard}
J. Hadamard (1865--1963),
{\sl Le\c cons de g\'eometrie \'el\'ementaire, 2 vols.,}
A. Colin, Paris, 1937.
\bibitem{Hammett}
D. Hammett, {\sl The Maltese Falcon,} Penguin, Middlesex, 1930.
\bibitem{Hayashi}
E. Hayashi, {\sl A reconstruction of the proof of Proposition~11
in Archimedes method,} Historia Sci. {\bf 3} (1994), 215--230.
\bibitem{Hardy}
G.H. Hardy (1877--1947),
{\sl A Mathematician's Apology,}
Cambridge University Press, New York, 1985.
\bibitem{Hartshorne}
R. Hartshorne, {\sl A Companion to Euclid, a course of geometry
based on Euclid's Elements and its modern descendents,}
AMS, Berkeley Center for Pure and Applied Mathematics, 1997.
\bibitem{Heath2}
T. Heath, {\sl A History of Greek Mathematics, vol.~II,}
Dover, New York, 1981.
%\bibitem{Heegner}
%K. Heegner, {\sl Diophantische Analysis und Modulfunktionen,}
%Math. Z. {\bf 56}, 227--253.
\bibitem{Heiberg1}
I.L.~Heiberg, {\sl Eine neue Archimedes--Handschrift,}
Hermes {\bf 42} (1907), p.~235.
\bibitem{Helmholtz}
H.L.F. von Helmholtz,
{\sl Helmholtz treatise on physiological optics,} translated from
the 3d German ed., edited by J.P.C.~Southall, Dover, New York, 1962.
\bibitem{Harvard}
D. Hughes-Hallett, A.M. Gleason et al., {\sl Calculus,}
John Wiley \& Sons, New York, 1994. Second edition, 1998.
\bibitem{Hutchins}
{\sl Great Books of the Western World,
vol.~11,} R.M.~Hutchins, editor, Encyclopaedia Britannica, Inc.,
Chicago, 1952.
\bibitem{NYT}
G. Johnson, {\sl The Big Question: Does the Universe Follow
Mathematical Law?}
The New York Times, February~10, 1998.
\bibitem{Katz}
V.J. Katz,
Review of ``Force and Geometry in Newton's Principia,'' by
F. de Gandt, American Math. Monthly {\bf 105} (1998), 386--392.
\bibitem{Klein}
F. Klein (1849--1925),
{\sl Elementary Mathematics from an Advanced Viewpoint,
2~vols,} Macmillan, New York, 1939.
\bibitem{Knorr0}
W.R. Knorr, {\sl The evolution of the Euclidean elements,}
Synthese Historical Library {\bf 15}, D.~Reidel,
Dordrecht--Boston, MA, 1975.
\bibitem{Knorr1}
W.R. Knorr, {\sl Archimedes and the Spirals: The Heuristic
Background,} Historia Math. {\bf 5} (1978), 43--75.
\bibitem{Knorr2}
W.R. Knorr, {\sl The Ancient Tradition of Geometric Problems,}
Birkh\"auser, Boston, 1986.
\bibitem{Knorr3}
W.R. Knorr, {\sl The method of indivisibles in ancient geometry,}
in ``Vita mathematica'' (Toronto, ON, 1992; Quebec City, PQ, 1992),
67--86, MAA Notes {\bf 40}, Math. Assoc. America, Washington, DC, 1996.
\bibitem{Lasky}
K. Lasky, {\sl The Librarian who Measured the Earth,}
Little, Brown \& Co., Boston 1994.
\bibitem{Lebesgue}
H. Lebesgue, {\sl Le\c cons sur les Constructions G\'eometriques,}
Gauthier--Villars, Paris, 1950. Reissued, Jacques Gabay,
Paris, 1987.
\bibitem{Moise}
E. Moise, {\sl Elementary Geometry from an Advanced
Viewpoint,} Addison--Wesley, Reading, MA, 1974.
%\bibitem{Morgan}
%F. Morgan, {\sl Alien mathematics and the first day of Spring,}
%Math Chat, March 3, 1998,
%{\tt www.csmonitor.com/yfiles/chat/chat030398.html}
%(The Christian Science Monitor, electronic edition).
\bibitem{Netz}
R. Netz, personal communication.
\bibitem{Nicomachus}
Nicomachus of Gerasa (circa 100~A.D.),
{\sl Introduction to Arithmetic,}
translated by M.L.~D'Ooge, in
\cite{Hutchins}. See \cite[Vol.~1, p.~101]{Thomas}.
\bibitem{Pappus}
Pappus, {\sl La Collection Math\'ematique,} traduit avec une
introduction et notes par P.~Ver~Ecke, Descl\'ee de Brouwer,
Paris, 1933.
\bibitem{Plutarch}
Plutarch, {\sl Lives, vol.~5,} translated by B.~Perrin,
Loeb Classical Library {\bf 87}, Harvard University Press,
Cambridge, MA, 1917. Also, translated by John~Dryden (1631--1700),
{\tt http://www.oed.com/plutarch.html.}
\bibitem{Rickey}
F. Rickey, {\sl The Archimedes manuscript,}
{\tt www.maa.org/news/sawit.html}.
\bibitem{Rickey2}
F. Rickey, {\sl The Archimedes palimpsest,}
{\tt forum.swarthmore.edu/epigone/math-history-list/mangzhunni}.
%\bibitem{Rosen}
%M.I. Rosen, {\sl Niels Hendrik Abel and equations of the fifth
%degree,} Amer. Math. Monthly {\bf 102} (1995), 495--505.
\bibitem{Sanchez}
F. Sanchez--Faba, ``Abraham Bar Hiyya (1070-1136) and
his "Libro de geometria," (Spanish) Gac. Math., I {\bf 32}
(1980) 101--115.
\bibitem{Selberg}
A. Selberg, {\sl Collected Works, 2 vols,} Springer Verlag, New York,
1989, 1991.
\bibitem{Simmons}
G.F. Simmons, {\sl Calculus with Analytic Geometry,}
McGraw Hill, New York, 1985.
\bibitem{Stark1}
H.M. Stark, {\sl On the ``gap'' in a theorem of Heegner,}
J. Number Theory {\bf 1} (1969), 16--27.
%\bibitem{Stark2}
%H.M. Stark, {\sl A historical note on complex quadratic fields
%with class--number one,} Proc. Amer. Math. Soc. {\bf 21}
%(1969), 254--255.
%\bibitem{Steiner}
%J. Steiner, {\sl Gesammelte Werke, 2 vols.,} Reimer, Berlin, 1882.
%\bibitem{Steiner2}
%J. Steiner, {\sl Geometrical Constructions with a Ruler Given
%a Fixed Circle with its Center,} Translated by M.E. Stark,
%Scripta Mathematica, New York 1950.
\bibitem{Swann} H. Swann, {\sl Commentary on rethinking rigor in
calculus: The role of the mean value theorem,} Amer. Math. Monthly
{\bf 104} (1997), 241--245.
\bibitem{Thomas}
I. Thomas, {\sl Greek Mathematical Works, 2 vol.,}
Loeb Classical Library {\bf 335}, {\bf 362},
Harvard University Press, Cambridge, MA, 1980.
\bibitem{Toussaint}
G. Toussaint, {\sl A new look at Euclid's second proposition,}
Math. Intelligencer {\bf 15} (1993), 12--23.
\bibitem{Vardi}
I. Vardi, {\sl A classical reeducation,} in preparation.
\bibitem{Vardi2}
I. Vardi, \begin{greek} ARQIMHDOUS PERI TWN ELIKWN EFODOS\end{greek},
in preparation.
\bibitem{Waerden}
B.L. van der Waerden, {\sl Science Awakening I,} Scholar's
Bookshelf Press, Kluwer Academic Publishers, Dordrecht,
The Netherlands, 1988.
\bibitem{Wiles}
A. Wiles, {\sl Modular elliptic curves and Fermat's last theorem,}
Ann. of Math. {\bf 141} (1995), 443--551.
\end{thebibliography}
\medskip
IHES\\
35 route de Chartres\\
91440 Bures--sur--Yvette\\
France\\
e--mail: {\tt ilan@ihes.fr},
website: {\tt www.ihes.fr/\~{}$\!$ilan/}
}
\psfig{figure=ancient_figure1.ps}
\psfig{figure=ancient_figure2.ps}
\psfig{figure=ancient_figure3.ps}
\psfig{figure=ancient_figure4.ps}
\psfig{figure=ancient_figure5.ps}
\psfig{figure=ancient_figure6.ps}
\psfig{figure=ancient_figure7.ps}
\psfig{figure=ancient_figure8.ps}
\end{document}