\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage[utf8]{inputenc}
\usepackage[greek.polutoniko,english]{babel}


\begin{document}

\section{Introduction}

Textbook presentations of mathematics itself and histories of the subject unfortunately tend to suffer from
some similar defects.  In most mathematics textbooks, everything is seemingly inevitably and tidily organized.  The actual process by which
humans discover new mathematics (even something new only to a student) rarely, if ever, comes through; the focus is often on acquiring cut-and-dried 
techniques and mastering well-known algorithms.  Analogously, until quite recently, histories 
have often presented aspects of the mathematical past as leading inevitably and tidily to our current understanding of the subject.  Modern 
concepts and techniques have been freely applied to interpret original sources; little effort has been made to meet that past on its own terms
or to understand the sometimes tortuous process by which mathematics has developed.  

I have long wondered whether a better 
understanding of the human side of mathematics as revealed through its history might provide tools to improve the teaching of 
mathematics and change the way the subject is perceived by non-experts.  
And so, because the Greek mathematical tradition has been so important for the way mathematics has developed in 
the West, for the past four years, I have been learning ancient Greek -- an intellectual adventure that I wish could be more common in
today's secondary and post-secondary education.\footnote{I think the mental discipline and concentration needed even to begin to tackle 
a language with such an involved grammar, such a long history, and such a huge vocabulary are salutary in and of themselves. 
I would say that mental discipline and concentration are comparable to the habits of mind needed to do research in mathematics,
even though there is not quite the same opportunity to exploit insight and creativity.  Greek is supremely logical in its way, but
many things must simply be committed to memory.  Nevertheless I have experienced immense pleasure along the way
and I would recommend such a course to anyone.}    My motivation for this was a desire to read works 
such as Euclid's {\it Elements} and the {\it Conics} of Apollonius  (or at least the portions of Apollonius that survive in Greek) 
in their original forms, guided by the recognition that all translations misrepresent their sources to some degree.  Moreover, 
the mathematics of the past often starts from completely different assumptions and uses different methods that may not be captured in modern
translations or expositions. Formally, I have been a special student in courses taught by several very kind 
colleagues in my home institution's well-regarded Classics department.  All of them, by the way, say that really mastering ancient 
Greek is the work of a lifetime and I would certainly not claim to have done that.  Yet I have read large chunks of those texts now and as a result 
I find my scholarly interests changing and my interest in the history of our discipline growing substantially.

In this essay I want to present a ``report from the field,'' so to speak, describing a recent encounter with two well-known passages,
one from one section of Plutarch's so-called {\it Moralia}, the compendium of occasional essays and miscellaneous writings that accompanies
the series of parallel {\it Lives} of illustrious Greeks and Romans in his immense output, and the other from his
{\it Life of Marcellus}.\footnote{This essay originated in an assignment
in Professor Thomas Martin's Plutarch seminar at Holy Cross in fall 2016 and I want to thank him for several very helpful comments
and suggestions.}  These passages touch on a crucial episode in the history of the Greek mathematics of the fourth century BCE
that stimulated mathematical research into the 19th century CE and
whose influence is still felt at one point in the undergraduate pure mathematics curriculum--the story of various approaches to 
the duplication of the cube.\footnote{This problem is usually grouped together
with two others, the quadrature of the circle (i.e. the problem of constructing a square or rectangle with
the same area as a given circle), and the trisection of a general angle. Wilbur Knorr examined this 
tradition of geometric problems in detail in [K1].  These passages from Plutarch have been seen 
by some (see [S] for a discussion) as a major contributing
factor to the later notion that in Greek geometric constructions, \emph{only} the
compass and straightedge were acceptable tools.  In fact constructions using
auxiliary curves of various sorts were developed for the quadrature of the circle
and the trisection of general angles as well.  Whether these constructions could be accomplished using only the Euclidean tools 
remained an open question until the work of P. Wantzel and others in the 19th century CE, completing an line of thought
initiated by Descartes.  It is now known that
\emph{none} of them is possible under those restrictive conditions.  Many 
undergraduate mathematics majors learn proofs of these facts in abstract algebra courses.}

One theme will be the way some key sources come from texts that have much wider cultural contexts and resonances. Sensitivity to 
the history, to the mathematics, and to the language is necessary to tease out their meanings.  Yet, historians of mathematics have often 
interpreted these sources using the mathematics of their own times and produced what we can now see are questionable conclusions.  
Unfortunately it is those sometimes anachronistic 
accounts that have been presented in mainstream histories of mathematics to which mathematicians who do not read Greek must 
turn to learn about that history.   At the same time, some classicists lack the mathematical background knowledge to appreciate that 
aspect of the content of these sources.   Another theme is that once one returns to the actual source material, questions with no easy 
answers often abound and it is amazing to see how much we still do not understand about critical junctures in the history of our subject.  
The tidy and inevitable picture of the development of mathematics disappears and we are left with a much more interesting, if 
ultimately somewhat inconclusive, story.  

Because this period and the associated questions
have been intensively studied since the 19th century CE, I cannot claim that this essay presents any new historical scholarship.  However,
I hope that it may prove useful to instructors and other readers who are interested in finding more nuanced accounts of 
some of the Greek work on the duplication of the cube than are available in some of the standard histories.

\section{The Plutarch Texts}

Plutarch of Chaeronea (ca. 45 - ca. 120 CE) was a Greek writing for a mixed Greek and Roman audience during the early empire. 
He himself records that he studied philosophy and mathematics in Athens and his writings reveal a strong connection with Platonic traditions.
We would call him an essayist and biographer, although most of 
his writing is more devoted to ethical lessons than to history, {\it per se}.  
The first passage I will discuss comes from a section of his {\it Moralia}  known as the {\it Quaestiones Convivales}, or ``Table Talk.''   
Each section of this work is presented as a record of  conversation at a {\it sumposion}, or drinking party, arranged by Plutarch for 
a group of guests.   Philosophical questions are always debated and it is amusing to see what Plutarch says about 
the rationale for this:  in ``... our entertainments we should use learned and philosophical discourse ...'' so that even if the guests become
drunk, ``... every thing that is brutish and outrageous in it [i.e. the drunkenness] is concealed ... ."\footnote{[QC], 716g, Book 8, Chapter 0, Section 2}  
In other words, to keep your next party from degenerating into a drunken brawl, have your guests converse about Plato!

In Book 8, Chapter 2, Section 1 (classicists refer to this via the so-called Stephanus page 718 ef, from
one of the first modern printed versions of the Greek text), Plutarch presents a conversation between
the grammarian Diogenianus and the Spartan Tyndares concerning the role of the study of geometry in Plato's thought. 
Diogenianus begins this phase of the conversation by raising the question why Plato asserted that ``God always 
geometrizes."  He also says he is not aware of any specific text where Plato said precisely that, though he 
thinks it sounds like something Plato would have said.  Tyndares replies that there is no great mystery there
and asks Diogenianus whether it was not true that Plato 
had written that geometry is ``... taking us away from the sensible and turning us back to the eternal nature 
we can perceive with our minds, whose contemplation is the goal of philosophy ... ."\footnote{In the text, I will present
my translations of the Greek.  Here:  ``\textgreek{>aposp\~wsan <hm\~as prosisqom'enous t\~h| a>isj'hsei ka`i >apostr'efousan
>ep`i t`hn noht`hn ka`i >a'idion f'usin, \~<hs j'ea t'elos >est`i filosof'ias ... }." All Greek quotations from the Plutarch passage are from the Loeb Classical
Library/Perseus online text of Plutarch's \emph{Quaestiones Convivales}, [QC].}  Tyndares is evidently thinking of passages like 
527b in Book VII of the {\it Republic}, where Plato has Socrates say in reference to geometry, ``... it is the knowledge of that which always is ... 
it would tend to draw the soul to truth, and would be productive of a philosophical attitude of mind, directing upward the faculties
that are now wrongly turned downward.''\footnote{English translation from [P], p. 758.}

After some elaboration of these points, Tyndares presents an interesting piece of evidence concerning
this aspect of Plato's thought: 
``Therefore even Plato himself strongly criticized Eudoxus, Archytas, and Menaechmus" (or possibly ``those around  Eudoxus, Archytas, and
Menaechmus'')  ``for attempting to reduce the duplication of the cube to tool-based and mechanical constructions ... .''\footnote{``\textgreek{d\'io ka`i 
Pl'atwn a>ut`os >em'emyato to`us per`i E>'udoxon ka`i >Arq'utan ka`i 
M'enaiqmon e>is >organik`as ka`i mhqanik`as kataskeu`as t`on to\~u stereo\~u diplasiasm`on >apage\~in >epiqeiro\~untas ... }''}
Tyndares continues in a somewhat technical vein, ``... just as though they were trying, in 
an unreasoning way,  to take two mean proportionals in continued proportion any way that they might ... .''\footnote{
``\textgreek{<'wsper peirwm'enous di' >al'ogou d'uo m'esas 
>an'alogon, \~<h| pare'ikoi, labe\~in ... }''  The \textgreek{di' >al'ogou} is hard to translate and may not even be 
what Plutarch originally wrote.  This specific phrase has a rather large number of textual issues
as evidenced by the variant readings discussed in the Loeb Classical Library/Perseus version of the
Plutarch text.  At least one editor has suggested that the whole phrase should be omitted from the text!}   
It is quite interesting that Tyndares seems to be assuming that all of his listeners would be 
familiar with this terminology and the episode in the history of geometry to which he is referring. 
Tyndares concludes his summary of Plato's criticism by claiming that in this way (i.e. by using mechanical procedures with
tools)  ``... the good of geometry is utterly destroyed and it falls back on things of the senses; it is neither carried above 
nor apprehends the eternal and immaterial forms, before which God is always God.''\footnote{``\textgreek{>ap'ollusjai 
g`ar >o'utw ka'i diafe'iresjai t`o gewmetr'ias >agaj`on a\~>ujis >ep`i t`a a>isjht`a palindromo'ushs ka`i m`h ferom'enhs >'anw 
mhd' >antilambanom'enhs t\~wn >a'idiwn ka`i >aswm'atwn e>ik'onwn pr`os a\~>isper >`wn <o je`os >ae`i je'os >esti.}''}
Tyndares is saying that Plato criticized the nature of the solutions proposed by Eudoxus, Archytas, 
and Menaechmus because they in effect subverted what he (i.e. Plato) saw as the true purpose of geometry: its \emph{raison
d'\^etre} was not merely to solve problems ``by any means necessary,''  but rather to lead the soul to the contemplation of 
eternal truth.\footnote{In his thought-provoking study of the Greek work on these construction problems, [K1], Wilbur 
Knorr has argued in effect that by this period Greek mathematics had distanced itself from the sort of 
philosophical or religious underpinning that Plutarch has Tyndares say Plato claimed for it and 
was already very close to a modern research program, in which the goal \emph{often is indeed} at to 
solve problems by whatever means are necessary, and then try to understand what methodological restrictions
might still allow a solution.  (See the discussions on pp. 39--41 and p. 88 in particular.) Needless to say, this view is 
not universal among historians of Greek mathematics.}

It is interesting to note that Plutarch gives a second, partially parallel account of this criticism in Chapter 14, Sections 5 and 6
of his {\it Life of Marcellus}, 
in the context of a discussion of the geometrical and mechanical work of Archimedes and the tradition that King Hiero 
of Syracuse persuaded him to take up mechanics to design engines of war in defence of his native city-state.  Marcellus was, 
of course, the commmander of the Roman forces in the siege of Syracuse in 212 BCE during which Archimedes was killed. 
In that passage, Plutarch says\footnote{(in the English translation by Bernadotte Perrin from [LM])}

\begin{quotation}
\baselineskip=10pt
``For the art of mechanics, now so celebrated and admired, was first originated by Eudoxus and Archytas, 
who embellished geometry with subtleties, and gave to problems incapable of proof by word and diagram, 
a support derived from mechanical illustrations that were patent to the senses.  For instance in solving
the problem of finding two mean proportional lines, a necessary requisite for many geometrical figures, 
both mathematicians had recourse to mechanical arrangements\footnote{``\textgreek{kataseku'as}'' -- ``constructions'' would
be another, perhaps better, translation here.}  adapting to their purposes certain intermediate
portions of curved lines and sections.\footnote{``\textgreek{mesogr'afous tin`as >ap`o kamp'ulwn ka`i tmhm'atwn
mejarm'ozontes}.''  I think a better translation here is ``adapting to their purposes mean proportionals found from
curved lines and sections."  The meaning of \textgreek{mesogr'afous} here is the same as in a passage from 
Eratosthenes that will be discussed below.  One can see the dilemma of a non-mathematician translating technical discussions!  In
the passage as a whole, one can also glimpse some of the less attractive aspects of Plato's thought.}
  But Plato was incensed at this, and inveighed against them as corrupters
and destroyers of the pure excellence of geometry, which thus turned her back upon the incorporeal things of 
abstract thought and descended to the things of sense, making use, moreover, of objects which required much
mean and manual labor.  For this reason, mechanics was made entirely distinct
from geometry, and being for a long time ignored by philosophers, came to be regarded as
one of the military arts.''
\end{quotation}

As we will see, these passages in Plutarch provide a 
fascinating, but also ultimately somewhat cryptic, sidelight on a key episode in the history of Greek mathematics.

\section{The Historical Background}

Eudoxus of Cnidus (409--356 BCE), Archytas of Tarentum (428--347 BCE), and Menaechmus of Alopeconnesus (380--320 BCE)
were three of the most accomplished Greek mathematicians active in the 4th century BCE.  Archytas is often 
identified as a Pythagorean and there are traditions that Eudoxus was a pupil of his and
Menaechmus was a pupil of Eudoxus.   All three were associated with Plato and his Academy in Athens in some way.\footnote{We
have much of this from sources such as Proclus, [P], pp. 54--56, though the fact that Proclus is writing roughly 800 years after this period
raises the question of how reliable his information is.}

As almost all mathematicians know, the \emph{duplication of the cube} was a geometrical problem 
asking for the construction of the side of a cube whose volume would be twice the volume of a given cube.  
Various traditions deal with the genesis of this problem.  One says that seeking
direction in order to stem the progress of a plague on their island (or perhaps political conflicts; 
different versions of the story differ on this point), the people of Delos consulted the oracle 
at Delphi, whereupon the Pythia replied that they must find a way to double the size of an 
cubical altar of Apollo.\footnote{A somewhat parallel story about King Minos seeking how to 
double the size of a tomb also appears in a letter of Eratosthenes to King Ptolemy III Euergetes
of Egypt that will be discussed below.  See [H], p. 245.}
When they were unable to do this themselves, the Delians supposedly consulted 
Plato and the geometers at his Academy to find the required geometric construction; 
for this reason the problem of the duplication of the cube is often called the ``Delian problem.''  
However,  this version of the story is almost certainly fanciful (at least as the \emph{origin} of the problem--the 
chronology seems to be wrong, for one thing, since as we will see presently there was work 
on the question somewhat before the time of Plato (428 -- 348 BCE)).  
Plutarch himself, in  another section of the {\it Moralia} called {\it The E at Delphi}, says that the 
underlying point of the story was that the god was commanding the Greeks to apply themselves 
more assiduously to geometry.\footnote{[DeE], Chapter 6, 386e.}   There is no doubt that the duplication
of the cube was one of a series of geometric construction problems that stimulated 
the development of Greek mathematics throughout the Classical period.

For a full understanding of our Plutarch passages, and of Plato's supposed objection to the work of Eudoxus, Archytas, and 
Menaechmus, we need to introduce an important piece of progress that had been made earlier and definitely before the 
time of Plato by Hippocrates of Chios (ca. 470--ca. 410  BCE).  None of Hippocrates' own writings have
survived and we know about the following only from sources such as fragments of a history 
of pre-Euclidean mathematics by Eudemus of Rhodes (ca. 370 -- ca. 300 BCE) preserved in other sources.    Given two 
line segments $AB$ and $GH$, we say line segments $CD$ and $EF$ are \emph{two mean proportionals in 
continued proportion}\footnote{In the first Plutarch passage above, this appears in the accusative as
\textgreek{d'uo m'esas >an'alogon}.  The \textgreek{>an'alogon} seems to be essentially equivalent
to the \textgreek{>an`a l'ogon} from the second passage, and that is conventionally translated as ``in continued proportion.''}
between $AB$ and $GH$ if their lengths satisfy:
\begin{equation}
\label{tmp}
\frac{AB}{CD} = \frac{CD}{EF} = \frac{EF}{GH}.
\end{equation}
 Hippocrates' contribution was the realization that if we start with 
$$GH = 2AB,$$
then the construction of two mean proportionals as in (\ref{tmp}) would \emph{solve the problem of the
duplication of the cube.}  The idea is straightforward:  If
$$\frac{AB}{CD} = \frac{CD}{EF} = \frac{EF}{2 AB},$$
then some simple algebra (which the Greeks would have emulated with parallel manipulation of proportions)
shows 
$$CD^3 = 2AB^3.$$
In other words, if $AB$ is the side of the original cube, then $CD$ is the side of the cube with 
twice the volume.  With this observation, Hippocrates did not give a full solution for the duplication of the cube,
but he did provide a way to attack the problem.  Almost all later work took his reduction to the construction
of the two mean proportionals as its starting point.  

\section{Do We Know What Eudoxus, Archytas, and Menaechmus Did?}

Plutarch does not include any discussion of what Eudoxus, Archytas, or Menaechmus actually 
did in their work on the duplication of the cube.  However, historical accounts
of the work on this problem including information about their approaches have survived in ancient
sources.  The most important is a much later commentary 
on Archimedes' {\it On the Sphere and the Cylinder} by Eutocius of Ascalon (ca. 480 -- ca. 540 CE), in which Eutocius
surveys a wide selection of different solutions to the problem of duplicating a cube by finding two mean proportionals, as
discussed in our Plutarch passages.\footnote{The occasion for the commentary
was the fact that Archimedes assumed the construction was possible in some way in the proof of the first
proposition in Book II of {\it On the Sphere and the Cylinder}, but he did not provide any explanation.  Because of the 
nearly 1000 years intervening between the time of the Platonic geometers and Eutocius' time, the caveat we made above 
about accepting evidence from Proclus' writings uncritically also applies to Eutocius' writings. The Greek original with 
a Latin translation is included in Volume III of J. Heiberg's 
\emph{Archimedis Opera Omnia}, [A].  A near-literal English translation is given by Netz in [N].}  
Eutocius includes in his account a purported letter to King Ptolemy III of Egypt by Eratosthenes of Cyrene (276 -- 194 BCE) with a summary
of earlier work and Eratosthenes' own, definitely mechanical and tool-based, solution making use of an instrument
dubbed the \emph{mesolabe}, or ``mean-taker.''\footnote{The purpose of the letter is essentially
to claim the superiority of Eratosthenes' tool-based mechanical method for practical use.  It was dismissed as a later forgery by some 19th and 
early 20th century historians, but more recently, the tide of opinion has seemingly changed and sources such as [K1] argue
that it should be accepted as authentic.}

I will discuss the main ideas behind what Eutocius says about the approaches of Archytas 
and Menaechmus and the ways this evidence has been interpreted.  The solution by Eudoxus is not
presented in detail by Eutocius because he believes his sources for it are corrupt.  Hence we
do not have enough information to draw any conclusions about how it connects with
what Plutarch writes.    My goals here are to shed some 
additional light on the content of the Plutarch passages and to show how different interpretations of
what Eutocius says have led to quite different understandings of this part of Greek geometry.  Some
of these seem more faithful to context of this work and some seem more anachronistic.  

To begin, we should discuss how  ``instrument-'' or 
``tool-based''\footnote{\textgreek{>organik'os}} and ``mechanical''\footnote{\textgreek{mhqanik'os}} or
might apply to geometric constructions.   On the face of it, ``instrument-based,'' 
or ``tool-based,'' is clearer.  For that adjective to apply, I believe some physical device such as the 
\emph{mesolabe} of Eratosthenes must be involved in the construction.   But even there, there is a point that is 
subtle for some modern readers of these works.  
The Greeks, even though they used physical straightedges to draw lines and physical compasses to 
draw circles while constructing diagrams, also considered those tools in \emph{idealized versions} that were constructs
of the mind and not dependent on the senses.  The first three postulates in Book I of the {\it Elements}
of Euclid (ca. 300 BCE) describe their uses and properties in abstract terms.  In particular, the idealized straightedge
can be used to draw lines, but not to measure distances; it has no distance scale like a modern ruler.  Moreover, the 
idealized Euclidean compass can be used to draw circles, but not to measure or transfer distances.  So we should
certainly not take criticisms such as the one ascribed to Plato here to refer to constructions that involve
the Euclidean tools. 

What counts as ``mechanical'' is not entirely clear either.  For Plutarch, from the evidence of the discussion in the 
\emph{Life of Marcellus}, it seems that the adjective ``mechanical'' referred to the use of mathematics to design and
emulate the motions of real-world
machines, and perhaps machines of war in particular.  It is possible that for him the adjectives ``tool-based'' 
and ``mechanical'' overlap in meaning to some degree.  Hence, I would suggest that another aspect that might make a geometrical construction
``mechanical,'' apart from the use of actual tools, is 
that it has some element of \emph{motion} (real or imagined) as in a real-world mechanical device.  Note that any sort of change 
over time in a figure would by itself seem to violate Plato's vision of the eternal and unchanging nature of the world of the forms.  

\section{The Work of Archytas}

In Eutocius' presentation, the account of the approach by Archytas is specifically attributed to Eudemus' history, 
which means what we have may actually be a commentary on a commentary.\footnote{The other sections of Eutocius' commentary
are not labeled in this way; they give the name of the author, and sometimes the title of the work from which
Eutocius is quoting.}  The solution
 is essentially based on a geometric configuration in which it can be
seen that two mean proportionals in continued proportion have been found.  Borrowing from [M], we will call these
\emph{Archytas configurations}.  One of these is shown in Figure~1.
\begin{figure}
\begin{center}
\includegraphics[width=2.5in]{Archytas1.png}
\caption{An \emph{Archytas configuration}}
\end{center}
\end{figure}
Here $AEB$ and $ADC$ are two semicircles tangent at $A$, and $BD$ is tangent to the 
smaller semicircle at $B$.  
It follows from some standard geometric facts that $\Delta BAE$, $\Delta CAD$, 
$\Delta DBE$, $\Delta CDB$ and $\Delta DAB$ are all similar.  This follows because
the angles $\angle AEB$ and $\angle ADC$ are inscribed in semicircles, hence right angles, and
hence the lines $\overleftrightarrow{EB}$ and $\overleftrightarrow{DC}$ are parallel.  From this we can see immediately that taking ratios
of longer sides to hypotenuses in three of these triangles,
$$\frac{AE}{AB}=\frac{AB}{AD}= \frac{AD}{AC}.$$
In other words, $AB$ and $AD$ are two mean proportionals in continued proportion 
between $AE$ and $AC$.  

But now, we must address the question of how such a configuration would be 
constructed given the lengths $AE < AC$.   A modern explanation might run as follows. The issue is that although we can 
always take the segment $AC$ as the diameter of  the larger semicircle, there is 
no direct way to construct the smaller semicircle, the perpendicular $BD$ to $AC$
and the point $E$ without some sort of continuity argument or approximation 
process.   Consider the situation in Figure 2.
\begin{figure}
\begin{center}
\includegraphics[width=2.5in]{Archytas2.png}
\caption{A failed attempted construction of an \emph{Archytas configuration}}
\end{center}
\end{figure}
Given the lengths $AE < AC$, the possible locations of the point $E$ lie on an arc
of the circle with center at $A$ and radius equal to a specified length.  One such arc is shown in blue
in Figure 2.  Through each point $E$ on that arc, there is exactly one semicircle
tangent at $A$ to the semicircle with diameter $AC$, shown in green in the figure.
The line through $A,E$ meets the outer semicircle at $D$ and $B$ is the foot of 
the perpendicular from $D$ to $AC$.  However, note that with this choice of $E$, 
$DB$ \emph{does not meet
the smaller semicircle at all}.  However, by rotating the segment $AE$ about $A$ and
increasing the angle $\angle CAE$,
we would eventually find that the corresponding $BD$ cut through the corresponding
smaller semicircle.  Hence there must be some point $E$ on the blue arc that
yields an Archytas configuration as in Figure 1, by continuity.  As we have
described it, a naive process
of finding that point might involve motion and exactly the sort of resort to ``eyeballing'' or 
use of the senses that Tyndares says Plato criticized in our passages from Plutarch!\footnote{We can 
also easily locate such a point using modern coordinate geometry, trigonometry, and numerical
root finding.  But needless to say, all of that is well beyond the scope of Greek mathematics.}

What Eutocius said that Archytas actually did here has been interpreted in a number of different ways
by different modern scholars.  One tradition known from influential sources such as
T.~L.~Heath's {\it A History of Greek Mathematics}, [H],
interprets Archytas' solution as a bold foray into solid geometry whereby a suitable
point like our $E$ in Figure 1 is found by the intersection of three different surfaces in three dimensions
(a cylinder, a cone and a degenerate semi-torus--the surface of revolution generated by rotating the semicircle with diameter $AC$
about its tangent line at $A$).\footnote{See for instance [H], pp. 246--249.  Heath characterizes
this solution as ``the most remarkable of all'' because of the sophisticated use of three-dimensional
geometry he sees in it.  Similarly, Knorr calls it a ``stunning \emph{tour de force} of stereometric insight'' in [K1], p. 50.}

Very recently, however,  a new interpretation based on a close reading of the Greek text of
Eutocius has appeared in the historical literature in [M].  As the author Masi\`a points out, it is not easy
to see all of the aspects of Heath's description in the actual text.  While a cylinder and a cone
are explicitly mentioned, the semi-torus surface of revolution is not.  Moreover, even there, the
cone and its properties are not really used in the proof; it seems to be included more for
the purposes of visualization.\footnote{[M], p. 203.}  Hence Heath's interpretation, while
certainly a correct way to describe the geometry, seems to be an anachronistic reading.

Instead, Masi\`a suggests that the argument can be understood in a fashion that seems much closer
to what we know about the state of geometry at the time of Archytas.  In a preliminary step, an 
Archytas configuration is found by starting from a semicircle and an inscribed triangle two of whose sides have the given
lengths $AE$ and $AC$.  A second copy of the semicircle and a moving inscribed triangle are rotated about $A$ until
an Archytas configuration is reached.  This motion is then emulated in three dimensions
by triangles in two perpendicular planes to produce a construction matching the Greek text of Eutocius very
closely.\footnote{See [M], pp. 188--193.  Masi\`a discusses several other possible
ways to interpret Archytas' solution in two or three dimensions and discusses other interpretations including
the one given in [K3], Chapter 5. He also points out that Heath's characterization
of this solution as the ``most remarkable'' does not seem to match the way the solution is presented by Eutocius.}  
In either our simple presentation, or the kinematic description
of the three surfaces in three dimensions, or the new reading of Archytas' construction from [M], there
is definitely an aspect of motion that seems to agree with Plato's reported 
characterization of the construction as ``mechanical.''\footnote{On the other hand, as Masi\`a points out on p. 197 of [M], 
the third person perfect imperative verb forms typically used in Greek to mark the steps of geometric constructions (e.g. 
\textgreek{gegr'afjw} -- ``let it have been drawn'')
seem to emphasize that the figure or diagram \emph{has been constructed} as a whole, and thus connote something static rather than 
something dynamic.  This convention seems in fact to agree perfectly with the Platonic conception of geometry that forms
the basis of the criticism in our Plutarch passages.  One can imagine Archytas responding to Plato's criticism by pointing
out that there is no actual motion involved!} How the adjective
``instrument-'' or ``tool-based'' might apply is not  as clear, although one could easily imagine
a device to carry out the planar version of the construction given in Figure 11 of [M].

\section{The Work of Menaechmus}

The approach attributed by Eutocius to Menaechmus is even more problematic although 
it was evidently extremely influential for the development of a key part of Greek geometry.
 This approach can be described
(very anachronistically) as follows.\footnote{This is essentially the presentation given 
in [H], pp. 252--255, although Heath does not use coordinate geometry explicitly in this way.}  
Given line segments of lengths $a,b$, finding the two mean proportionals
in continued proportion means finding $x,y$ to satisfy:
\begin{equation}
\label{Men}
\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.
\end{equation}
Hence, cross-multiplying and interpreting the resulting equations via coordinate geometry,
we see the solution will come from the point of intersection of the parabola $ay = x^2$ 
and the hyperbola $xy = ab$, or one of the points of intersection of the two parabolas $ay= x^2$ and
$bx = y^2$.  A related piece of evidence is the epigram of Eratosthenes on the 
duplication of the cube that concludes the letter to Ptolemy III mentioned above.
This includes the direction ``neither seek to cut the 
cone in the triads of Menaechmus'' to obtain a solution.\footnote{``\textgreek{mhd`e Menaiqme'ious kwnot'omein tri'adas diz'hai ... .},'' 
[A], p. 112.   (The present middle indicative \textgreek{diz'hai} should probably be the aorist middle subjunctive \textgreek{diz'hsh|} and other
sources correct it that way.)
See for instance [H], p. 246.  Note that
Eratosthenes was active significantly later than Menaechmus himself and the initial steps in the
elaboration of the theory of conic sections would have intervened.  Exactly what this phrase
means is also not entirely clear.  Some writers have seen in the ``triads'' the division of conic
sections into the three classes of ellipses, parabolas, and hyperbolas.  However to the extent
that it applies to Menaechmus' work at all, this is surely
anachronistic; it seems more likely that the ``triads'' refer essentially 
to the three curves obtained from the 
proportionality relations in (\ref{Men}).}

For these reasons, Menaechmus has often been credited with
initiating the study of the \emph{conic sections}, later taken up and elaborated by Euclid, 
Archimedes, Apollonius, and other Greek mathematicians.   However, it must be said that
although this claim has been repeated in a number of standard histories of mathematics, including
[H] and [BM]\footnote{[H], pp. 251--255; [BM], pp. 84--87.}, the
case that Menaechmus did any more than to stimulate those later developments on 
conic sections with his work
on the Delian problem is flimsy at best.  It rests more on imaginative reconstructions
of his ``probable methods'' produced by Zeuthen, Coolidge, and Heath in the 19th century  than on any 
direct evidence.\footnote{My thinking on this has been strongly influenced by [K2] and the discussion
in [K1], pp. 63--69.}


None of Menaechmus' own writings have survived and (even more suspiciously)
the discussion of his work in Eutocius uses the terminology for conic sections 
introduced much after the time of Menaechmus himself by 
Apollonius of Perga (262--190 BCE).\footnote{Since that terminology seems to have been
developed by analogy with constructions in the application of areas, there is no doubt that
some connection between Menaechmus and the later theory of conics exists. The point I wish
to make here is that it seems anachronistic to attribute this discussion, {\it in the form
given by Eutocius}, to Menaechmus himself.} Earlier terminology, according to which parabolas
are ``sections of right-angled cones'' (by planes perpendicular to one of the generating
lines of the cone) and hyperbolas are ``sections of obtuse-angled cones'' is preserved in such 
works as Archimedes'  {\it Quadrature of the Parabola}. 
This is sometimes used to infer the way Menaechmus may have approached the definition
of the conics as well as the way that theory may have been presented in the lost {\it Conics} of Euclid.  
On the other hand, in [K1], pp. 63--69, Knorr presents an alternative conjectural reconstruction
of Menaechmus' work that does not rely on curves derived as sections of cones.  His version
is also at least plausible.  In any case, it seems likely that either a source Eutocius consulted
or Eutocius himself reworked Menaechmus' presentation in the light of later
developments.  

Unfortunately, with our fragmentary knowledge from the surviving ancient sources, 
we cannot really be sure about any of this.  I would venture, though, that attributing a full-blown theory of 
conic sections (that is, as plane sections of cones) to Menaechmus may be 
yet another instance of the sort of {\it conceptual anachronism} that unfortunately abounds in 
conventional and accepted histories of mathematics.\footnote{This may in fact apply just
as much to Eutocius as to modern historians of mathematics.  In my opinion, mathematicians often
make bad historians.  They don't always distinguish between logically equivalent forms of 
statements and they tend to attribute their own understanding of those statements to mathematicians of the past.}

One of the slightly mysterious aspects of the Platonic criticism recounted in our 
Plutarch passages is how the adjectives ``mechanical'' or ``instrument-'' or ``tool-based''
might apply to what is attributed to Menaechmus by Eutocius.  It is true that the conic sections apart
from the circle cannot be constructed as whole curves using only the Euclidean tools and other
sorts of devices would be needed to produce them.  Interestingly enough, along these lines,
Eutocius' discussion does include a final comment that ``the parabola
is drawn by the compass invented by our teacher the mechanician Isidore of Miletus ... ."\footnote{``\textgreek{gr'afetai d`e <h parabol`h di`a to\~u
e<urej'entos diab'htou t\~w| Milhs'i\~w mhqnik\~w| >Isid'wrw| t\~w| >hmet'erw| didask'al\~w|} ... ", [A], p. 98.}
Isidore (442--537 CE) was an architect, one of the designers of the {\it Hagia Sophia} in Constantinople, 
and thus this note is surely an interpolation from the general period of Eutocius himself, not 
a part of the older source Eutocius was using to produce this section of his commentary.\footnote{See [N], p. 290, note 130.}

\section{Plato's Solution?}

In a final, decidedly odd, aspect of this story, Eutocius also gives a construction of the two mean proportionals 
in continued proportion that he ascribes to Plato himself.
But that is one of the most mechanical and tool-based of all the solutions he 
describes in that it requires the use of a frame something like two ``t-squares'' joined along one edge.\footnote{The 
text discussion is accompanied by a rare perspective drawing of the device.}
The configuration containing the two mean proportionals is found by maneuvering
the device until one point is found to coincide with one endpoint of one of the given segments.\footnote{There
is a good diagram of this on page 58 of [K1].}
 A sort of {\it neusis}, or limiting process, requring input from
the senses of the geometer is the crucial component.  As Knorr says, ``one is astounded at the 
flexibility of the traditions which on the one hand attribute such a mechanism to Plato, yet on the other hand
portray him as the defender of the purity of geometry and the sharp critic of his colleagues for their
use of mechanical procedures in geometric studies.''\footnote{[K1], p. 59.}

\section{Conclusions}

Our study of the passages in question shows that 
Plutarch has seemingly preserved a largely accurate picture of Plato's thinking, certainly more
accurate than some of the traditions preserved in Eutocius' commentary.  But from what we know of 
the work of Archytas and Menaechmus and from the later work of Archimedes, Apollonius and others, I would argue that if 
something like Plato's criticism of the geometers in his circle actually happened at this point in history,  then its effect on Greek 
mathematics was rather minimal.  

An openness to mechanical techniques can already be seen for instance in the description of the \emph{quadratrix} curve ascribed
to Hippias of Elis (late 5th century CE) and used in the period we have considered
in solutions of the angle trisection and circle quadrature problems.  We often find scholars of the Hellenistic and later
periods pursuing both mathematical and mechanical work, sometimes even in combination. Celebrated examples of this trend include Archimedes'
work on spirals, a portion of his \emph{Quadrature of the Parabola}, and most strikingly his \emph{Method
of Mechanical Theorems}, which presents a somewhat systematic procedure, based on mechanics, to discover geometric
area and volume mensuration results.\footnote{Archimedes uses dissection procedures akin to the subdivisions used in modern integral calculus, combined
with an idealized balance beam.  Most modern mathematicians would probably even be happy to consider his arguments as complete proofs, although
Archimedes himself had scruples about that point.}  Even later, Heron of Alexandria (ca. 10 -- ca. 70 CE)
gives another different solution of the problem of the two mean proportionals in his \textgreek{Belopoi\'ika}, a treatise on the design 
of siege engines and artillery(!)  

While it drew on philosophy 
for its norms of logical rigor, I would agree with Knorr that mathematics had in essence emerged as an independent subject in its own right by
the time of Eudoxus, Archytas, and Menaechmus.   Plutarch was, by training and inclination, a Platonist and this
by itself sufficiently explains his interest in preserving traditions about Plato's thinking about mathematics and
his criticism of the geometers in his circle.  Yet it is
doubtful that Plato's ideas about the proper methods or goals of mathematics carried much real weight for many of the actual practitioners 
of the subject.
\vfill\eject
\noindent
{\it References}
\begin{enumerate}
\item[]\hskip-20pt[A] {\it Archimedis Opera Omnia}, volume III, ed. J. Heiberg, Leipzig: Teubner, 1881.
\item[]\hskip-20pt [BM] Boyer, C. and Merzbach, U. {\it A History of Mathematics}, 3rd ed. New York: Wiley, 2011.
\item[]\hskip-20pt [DeE] Plutarch, {\it De E apud Delphi}, at\\ 
{\tt http.data.perseus.org/texts:um:cts:greekLit:tlg0007.tlg090.\\
persus-grc1}, referenced 12/30/2016.
\item[]\hskip-20pt [H] Heath, T.L., {\it A History of Greek Mathematics}, v. I, New York: Dover, 1981.
\item[]\hskip-20pt [K1] Knorr, W. {\it The Ancient Tradition of Geometric Problems}, New York: Dover, 1993.
\item[]\hskip-20pt [K2] Knorr, W. ``Observations on the Early History of Conics,'' Centaurus {\bf 26} (1982), 1--24.
\item[]\hskip-20pt [K3] Knorr, W.  {\it Textual Studies in Ancient and Medieval Geometry}, Boston: Birkhauser, 1989.
\item[]\hskip-20pt [LM] Plutarch, {\it Vita Marcelli}, at\\
{\tt http.data.perseus.org/texts:um:cts:greekLit:tlg0007.tlg022.\\
persus-grc1}, referenced 12/30/2016.
\item[]\hskip-20pt [M] Masi\`a, R., ``A new reading of Archytas' doubling of the cube and its implications,''
Arch. Hist. Exact Sci. {\bf 70} (2016), 175--204.
\item[]\hskip-20pt[N], Netz, R. {\it The Works of Archimedes, Volume 1:  The Two Books On the Sphere and the Cylinder}, Cambridge: Cambridge University 
Press, 2004.  
\item[]\hskip-20pt [P] Plato, {\it The Collected Dialogues}, Hamilton, E. and Cairns, H. eds.  Bollingen Series
LXXI, Princeton:  Princeton University Press, 1961.
\item[]\hskip-20pt [Pr] Proclus, {\it A Commentary on the First Book of Euclid's {\sl Elements}}, Morrow, G. translator,
Princeton:  Princeton University Press, 1970.
\item[]\hskip-20pt [QC] Plutarch, {\it Quaestiones Convivales}, at\\
{\tt http://data.perseus.org/texts:um:cts:greekLit:tlg0007.tlg112.\\
perseus-grc1}, referenced 12/30/2016.
\item[]\hskip-20pt [S]  Steele, A.D., ``\"Uber die Rolle von Zirkel und Lineal in der Griechischen Mathematik,'' Quellen und
Studien zur Geschichte der Mathematik, Astronomie, und Physik, Abteilung B--Studien, {\bf 3} (1936), 287--369.
\end{enumerate}
\end{document}


