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\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, 
<'wsper peirwm'enous di' >al'ogou}\footnote{variant readings: \textgreek{d'iqa l'ogou}, \textgreek{dial'ogou}} \textgreek{d'uo m'esas >an'alogon, \~<h| pare'ikoi, labe\~in,
>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.}
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\rightline{\textgreek{Sumposiak'a} ({\it Quaestiones Convivales}):  718 e-f (Book 8, Chapter 2, Section 1)}
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``Therefore even Plato himself harshly criticized  Eudoxus, Archytas, and Menaechmus for attempting 
to reduce the \emph{duplication of the cube} to mechanical constructions with instruments, just as though they were trying, in 
an unreasoning way,  to take two mean proportionals in continued proportion any way that they might; in this way the 
good of geometry is utterly destroyed and it falls back on the senses; it is not carried above to apprehend the eternal and 
immaterial forms, before which God is always God.''
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Two line segments $CD$ and $EF$ are \emph{mean proportionals in continued proportion} between $AB$ and $GH$ if
$$\frac{AB}{CD} = \frac{CD}{EF} = \frac{EF}{GH}$$
Hippocrates of Chios (ca. 470 -- ca. 410 BCE):  If, in addition, $GH = 2AB$, then 
$$AB\cdot EF = CD^2 \quad {\rm and} \quad 2AB^2 = CD\cdot EF,$$
so 
$$2AB^3 = CD^3.$$


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