\magnification=\magstep1
\noindent
{\it Translation from Hankel, H., Zur Geschichte der Mathematik in Alterthum und Mittelalter}
\bigskip
\noindent
Starting on page 155:
\bigskip
[Note:  This comes after a discussion of several of the ``mechanical'' constructions involved
in solutions of the problem of the duplication of the cube.]
\bigskip
If these methods were to be applied practically, then instruments must have been found that would 
let each curve be described mechanically.  It is very possible that, as is reported to us [footnote
refering to Plutarch, Quaest. conviv. VIII, q. 2, c. 1  and Vit. Marcelli, c. 14, \S 5.],
Plato, although he himself is supposed to have found an instrument for the solution of the 
problem in practice, disapproved of such instrumental and mechanical constructions,  saying
``because in this way the merit of geometry is spoiled; they [mechanical constructions] lead back to the
standpoint of the senses, instead of leading to higher things and dealing with eternal and 
immaterial images.'' For, thoroughly discounting practice, Plato also speaks of 
mathematicians:  ``They speak in a very ridiculous and restricted manner.  For they 
talk as if they are actually doing something when they say `squaring' or `applying'
or `adding,' while the whole subject is [actually] pursued purely for the sake of knowledge."

Viewed from this position, the great philosophers had correct convictions in this rejection
of mechanical constructions.  The problems that can be solved with only the compass
and straightedge form a limited and known class, and it is of the greatest importance, in 
embarking on each new problem, to determine whether it is solvable with those instruments
or not.  This is the same as in algebra where one asks whether an equation can be
solved using square roots or not.  If the introduction of arbitrarily many instruments
were allowed, then many important researches in this direction would not have been
undertaken.  We may then thank Plato for the restriction to these instruments, so
important for geometry.  The others, announced by their discoverers with much fanfare,
are today forgotten, because they lost all higher scientific importance.
\bye
