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Mathematics seems to endow one with something like a new sense.
In N. Rose (ed.), Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
The Classic Greek Ladder and Newton's Method
I am indebted to Professor Joseph Zund for introducing me to Greek ladders several years ago, and I thank Professors David Pengelley and Fred Richman for their very useful comments. The material in this article was presented at the MAA Southwestern Annual Conference, Western New Mexico University, Silver City, NM, April 3-4, 2009.
Robert J. Wisner is Professor Emeritus in the Department of Mathematical Sciences at New Mexico State University. He was founding editor of SIAM Review, the only one of the 14 journals of the Society for Industrial and Applied Mathematics (SIAM) designed to appeal to all of its members. Wisner also has authored or co-authored numerous K-12 textbooks for Scott Foresman and Company; was Consulting Editor for Mathematics for Brooks/Cole for over 25 years; and recently co-authored a series of interactive pre-calculus textbooks, available on CD from Hardy Calculus.
 Leonard Eugene Dickson, History of the Theory of Numbers, vol. II, Chelsea Publishing Company, New York, 1919, p. 341.
 John J. O'Connor and Edmund F. Robertson, “Roger Cotes,” MacTutor History of Mathematics Archive, http://www-history.mcs.st-andrews.ac.uk/Biographies/Cotes.html
Wisner, Robert J., "The Classic Greek Ladder and Newton's Method," Loci (August 2009), DOI: 10.4169/loci003330
MY COMMENTS ON THIS ARTICLE
I read with big interest this article of Dr. Robert J. Wisner (also the article on the same subject ÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬Ãâ¦Ã¢â¬ÅA disquisition of the square root of threeÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬ÃÂ¯ÃÂ¿ÃÂ½). It is very rewarding for a lover of mathematics as I am to find in Convergence article always very clear and very appealing. The only point that was not clear to me in this article was relevant to the ÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬Ãâ¦Ã¢â¬Årating systemÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬ÃÂ¯ÃÂ¿ÃÂ½. I understand from a personal E-mail interchanged with Prof. Wisner that in Diophantine approximation the goal is to get a good approximation with a minimal denominator, but following this line for sqrt(3) we can have these approximations: 5/3 with a rating of (-2,1) and a value 5/3=1,66666666666667 Or 19/11 with a rating of (-2,2) and a value 19/11 = 1,72727272727273 But the real value of sqrt(3) is 1,732051 therefore 5/3 is far from the value Following the explanation ÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬Ãâ¦Ã¢â¬Åin Diophantine approximation the goal is to get a good approximation with a minimal denominatorÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬ÃÂ¯ÃÂ¿ÃÂ½ there is no need for proceed along the ÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬Ãâ¦Ã¢â¬Ågreek ladderÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬ÃÂ¯ÃÂ¿ÃÂ½ because the approximation with a minimal denominator is always the first rung I like to share also a my explanation of the formula reported for the ÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬Ãâ¦Ã¢â¬Ågreek ladderÃÆÃÂ¢ÃÂ¢Ã¢â¬Å¡ÃÂ¬ÃÂ¯ÃÂ¿ÃÂ½, always said that is not known how was found For me this was the way If we say that b/a = sqrt(N) we have b^2/a^2 = N if we add to both said b/a we have b/a + b^2/a^2 = N + b/a b/a ( 1+b/a) = N + b/a b/a = [N+ b/a] / (1+b/a) and with some manipulation b/a = (a *N +b) /(a+b) from which bn = an-1 * N + bn-1 an = an-1 + bn-1 Best reagards Luigi Rivara