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Topic: Duplication of the Cube.
Replies: 0

 Domingo Gomez Morin Posts: 215 Registered: 12/3/04
Duplication of the Cube.
Posted: Oct 1, 1994 6:50 PM

An article entitled "New Elements For the Irrational Numbers" has been
published this month by :
"The Journal of Transfigural Mathematics". Berlin, Germany.
http://www.st.rim.or.jp/~shinichi/jtfm/

I think it could be of some interest to all the experts on the history
and the fundaments of mathematics.
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Abstract of the article :
"It is often claimed that by agency of algebra a modern man could solve
the problems no ancient Greek could do. The present work shows that
early mathematicians certainly had at hand an extremely simple operation
for solving problems which would imply the solution of any algebraic
equation, as for example the well known problem on the duplication of
the cube. It is presented the 'Rational Process' an iterative procedure
for finding rational approximations to the N-th. root of any positive
number by agency of the 'Mediant' which will be called here as the
'Rational Mean'. This method could have been easily implemented since
ancient times, mainly, because it only involves sums. So it is directed
to all those mathematicians interested in the historical attempts for
solving algebraic equations and deciphering the irrational numbers, in
this way, a brief introduction on these subjects is also included. For
the sake of clarity the examples are focused to find the cube root of 2,
the Golden Section and the transcendental number e."
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As all we know, the 'Mediant' is an extremely simple operation which
rules the generation of the 'Farey fractions' and the convergents in the
simple continued fractions.
It is astonishing this so simple method hasn't been used before the sway
of analytic geometry!!!.

Domingo GÃÂ³mez MorÃÂ­n.