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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.  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 Nth. 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."  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 GÃÂ³mez MorÃÂn. Email address : dgomezm@etheron.net



