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Topic: Lucas, irrational number, root-solving, Bernoulli Newton Halley Householder
Replies: 0

 arithmeticae Posts: 183 From: Venezuela Registered: 9/17/06
Lucas, irrational number, root-solving, Bernoulli Newton Halley Householder
Posted: Mar 6, 2009 11:33 AM

In the Introduction to his Théorie des Nombres Edouard Lucas said:

*************************************************
?Les ouvrages de Brahmegupta et de Bhascara Acharya donent la manière de déduire,
De?une soule solution, toutes les autres solutions entières d?une equation
indéterminée du second degree à deux inconnues, et cette analyse, quenous
attribuons á Euler et à Lagrange, Était connue aux Indes depuis plus de dix siècles¡
Mais nous devons rappeler surtout l?admir/able et rapide procédé d?extraction
de la racine carrée que nous expliquerons sur 2^(1/2) [?cut] On arrive ainsi
rapidement à deux suites de nombres:
les moyennes arithmétiques et les moyennes harmoniques[?cut]?
*************************************************

Lucas was describing the ancient Indian method (quadratic convergence)
for computing the square root of 2 by agency of arithmetic and harmonic means,
after that Lucas also wrote:

*************************************************
?Nous avons cherché pendant longtemps l?extension de ce procédé
aux raciness cubique, biquadratique, etc.; le lecteur en trouvera
les premières notions dans [?cut]?
*************************************************

In fact, Edouard Lucas was quite right, since ancient times
there were no generalizations of such Indian root-solving method (quadratic convergence, Newton?s method),
that?s why he recognized in his book to have worked so hard to find
an __arithmetical__ generalization in a similar way many others
unsuccessfully tried in past times.
Yes, all of them failed to do so, and frankly the complicated generalization
that Edouard Lucas finally developed by agency of the Arithmetic Mean
was so cumbersome to be used by hand, and as a matter of fact
it should not be considered as a true generalization.
Indeed, as Edourd Lucas asserted, all of them seem to have worked so hard on this matter
since Babylonian times, so it is just striking and befuddling to realize that,
now, to the light of the general Rational Mean concept, a new arithmetical
high-order generalization on such ancient Indian method
(and also on Bernoulli?s, Edouard Lucas?s, Newton, Halley?s and Householder?s methods)
have been developed as can be seen at the webpage:

http://mipagina.cantv.net/arithmetic/roots.htm

Links, comments, books, papers, and discussions on these new methods can be found at:

http://mipagina.cantv.net/arithmetic/

To self- disgrace of some few self-called experts on numerical algorithms
and the History of Mathematics, these new methods have not precedents
all through the very long history of root-solving. You can identify
those egotistic and funny self-called ?experts? by looking
at their ranting-raving posts in a futile attempt to hide that there are no trace
of any single precedent on these new methods.

As a matter of fact, before these methods were to be published
in these forums few time ago, there were lots of postings with questions
about square and cube roots, however, since the new methods were published there
have not been any other single post on the issue in this and other forums.

I understand all this represents a setback for some few egotistic self-called experts
on the History of Mathematics and numerical algorithms, but they have no way for fleeing away,
these new extremely simple methods came to stay for so long and will find their way
all through the math literature.
Edouard Lucas despite being a genius was enough modest to admit that he was working
so hard because there were no true general and natural __arithmetical__ root-solving methods,
and this tells so much about him not only as a true genius but as a rational human being.