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Plato's Criticism of Geometers of His Time\\
From Plutarch, {\it Quaestiones Convivales}, Book 8, Chapter 2, Section 1\\
Text for Oral Presentation
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\leftline{John Little}
\leftline{GREK 338 -- Plutarch Seminar}
\leftline{Prof. Martin}
\leftline{December 6, 2016}
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The Plutarch passage I will discuss comes from a section of the {\it Moralia} (\textgreek{>Hjik'a}) known as the 
\textgreek{Sumposiak'a}, or ``Table Talk.''   Each section of this work is presented as a record of  
conversation at a \textgreek{sump'osion} or drinking party arranged by Plutarch for a group of guests.   
Each of these parties involved consideration of some philosophical question 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. drunkenness!] is concealed ... ."  In other words,
to keep your next party from degenerating into a drunken brawl, have your guests converse about Plato!

During a celebration of Plato's birthday, Plutarch presents a conversation
on the role of geometry in Plato's thought.  One of the guests
begins by asking why Plato asserted that ``God is always doing geometry."    
A second guest replies that Plato wrote that geometry 
``takes us away from the sensible and turns us back to the eternal nature we
can perceive with our minds, whose contemplation is the goal of philosophy... .''
He then presents this interesting piece of evidence in our Greek text:
\vskip 10pt
\noindent
``\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 ... .}''
\noindent
``Therefore even Plato himself found fault with  Eudoxus, Archytas, and Menaechmus for attempting 
to reduce the \emph{duplication of the cube} to mechanical constructions with instruments  ... . ''  
\vskip 10pt
Now, the \emph{duplication of the cube} 
asked for the construction of the side of a cube whose volume would be twice the volume of a given cube.  This problem
came to be associated with a reputed pronouncement of the oracle at Delphi to the people of the island of Delos.  They were
instructed to construct an altar twice as large as an existing one to stop the progress of a plague on their island.   Elsewhere
in the {\it Moralia}, Plutarch says the oracle's agenda was actually to encourage the Greeks to pursue the study of geometry
more diligently.  If so, the commandment was definitely successful--this was one of a series of construction problems that stimulated 
much of the development of Greek geometry
through the Classical period.    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 being associated with Plato and his Academy in 
Athens in some way.

\vskip 10pt
\noindent
The second speaker continues in a surprisingly technical vein, \\
``\textgreek{<'wsper peirwm'enous di' >al'ogou d'uo m'esas 
>an'alogon, \~<h| pare'ikoi, labe\~in ... }''.

\noindent
This phrase has caused much perplexity (one editor proposes removing it entirely) and many variant readings.
As mathematics, though, this is a key point:  ``just as though they were trying, in 
an unreasoning way,  to take two mean proportionals in continued proportion any way that they might ... ''.  
[In the Plutarch passage, the ``two mean proportionals'' appears in the accusative as \textgreek{d'uo m'esas 
>an'alogon}.  The \textgreek{>an'alogon} seems to be essentially equivalent to \textgreek{>an`a l'ogon} -- ``in (continued)
proportion.'']  It is interesting that the speaker seems to be assuming 
 that all of his listeners will be familiar with this terminology and a key episode in the history of Greek geometry.
\vskip 10pt

A key piece of progress on the 
duplication of the cube was made somewhat before the 
time of Plato by Hippocrates of Chios (ca. 470--ca. 410  BCE).  Given two 
line segments $AB$ and $GH$, we say line segments $CD$ and $EF$ are \emph{two mean proportionals 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}
If in addition $GH = 2 AB$,
then some simple algebra [sketched on the handout] 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.  This was not a full solution, but it did provide a way to attack the problem and almost 
all later work took this as its starting point.  

\vskip 10pt
The speaker concludes his summary of Plato's criticism by claiming that [solving the problem that way]
 ``\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 e>ik'onwn pr`os a\~>isper >`wn <o je`os >ae`i je'os >esti.}"

\noindent
``... in this way the good of geometry is utterly destroyed and it falls back on the senses;
it is neither carried above, nor apprehends the eternal and immaterial images (forms), before which God is always God.''
That is, by their nature, the solutions proposed by Eudoxus, Archytas, 
and Menaechmus subverted  what Plato saw as the true purpose of geometry, which was not merely to solve 
problems ``by any means necessary,''  but rather to lead the soul to the contemplation of eternal truth.

Plutarch does not include any discussion of what Eudoxus, Archytas, or Menaechmus actually did.  However, such information  
has survived in an ancient source:  a much later commentary on Archimedes' {\it On the Sphere and Cylinder} by Eutocius [of Ascalon (ca. 480 -- ca. 540 CE)].
It's interesting (but also very subtle) to try to see to what extent the adjectives \textgreek{mhqnik'os} or \textgreek{>organik'os} from Plato's
supposed criticism apply.
If anyone is interested in the details, I would be happy to share a fuller write-up.

Here's the briefest of summaries: On the face of it, \textgreek{>organik'os}, in the sense of instrument-based, or 
tool-based, is clearer.  For  \textgreek{>organik'os} to apply, I believe some physical device must be involved
in the construction.   But there is a subtle point.  Even though
they certainly used physical straightedges to draw lines and physical compasses to draw circles while
constructing diagrams, Greek geometers considered those tools in \emph{idealized versions} (e.g. in 
the first three Postulates in Book I of Euclid's {\it Elements}) as purely mental constructs.  
Plato probably would have had no ``beef'' with the use of those tools.  Eutocius discusses much more elaborate
physical devices for this particular problem, but not in connection with Archytas or Menaechmus(!) 
What ``counts'' as \textgreek{mhqanik'os}
is unfortuately even less clear.  One possibility is a construction
that has some element of \emph{physical or imagined motion}.  Change over time in a figure
would seemingly violate Plato's vision of the eternal and unchanging nature of the world of the forms.
Archytas' solution definitely qualifies as \textgreek{mhqanik'os} in that sense.  

In conclusion, we can say that
Plutarch has seemingly preserved a debate on mathematical methodology that is at least
somewhat consistent with what we know from other sources.  But 
from the work of Archytas and Menaechmus and then Archimedes, 
Apollonius and others, if something like Plato's criticism of the geometers in his circle
actually happened, then its effect was minimal.  Mathematics had emerged from philosophy 
as a subject in its own right, mechanics was entering its mainstream,  
and Plato's ideas about the proper methods or goals of geometry were hardly the last word.  
\end{document}


I will present the main ideas behind what Eutocius says about the approach of Archytas in order to shed some 
additional light on the content of the Plutarch passage.

\begin{figure}
\begin{center}
\includegraphics[width=2.5in]{Archytas1.png}
\caption{An \emph{Archytas configuration}}
\end{center}
\end{figure}

The approach by Archytas is essentially based on a geometric configuration in which it can be
seen that two mean proportionals in continued proportion have been found.  We will call these
\emph{Archytas configurations} for simplicity.  One of these is shown in Figure~1.
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 $EB$ and $DC$ 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 $AE < AC$.  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.  A modern explanation might run as follows.   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}.  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.  However, as we have
described it, a naive process
of finding that point might involve exactly the sort of resort to ``eyeballing'' or 
use of the senses that Tyndares says Plato criticized in our passage from Plutarch!

What Archytas actually did here has been interpreted in a number of different ways
by different modern readers.  One tradition for instance 
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 the surface of revolution generated by rotating the semicircle with diameter $AC$
about the tangent line at $A$).  
Very recently, a new interpretation has been presented whereby the
Archytas configuration is found by rotating one copy of the semicircle around $A$ in the plane while
keeping the other one fixed.  In either our simple presentation or the one from [M], there
is definitely a \emph{mechanical} aspect that seems to agree well with Plato's reported 
characterization of the construction.  Either way, Archytas is clearly willing to allow the use of the senses 
of the geometer and/or mechanical operations in this construction.


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 by far 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. 
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.
 Again a sort of \textgreek{ne\~usis} 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.''

