ArchimedesCalculus 2008-07-14 09:44:18 2009-10-12 20:42:10 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The calculus of Archimedes has been passed down to the modern era within one text, an erasable parchment (palimpsest). The information was not recorded in Archimedes' handwriting. Worse, the parchment's calculus information had been copied over hundreds of years, and erased in 1,100 AD by Byzantine priests. Byzantines used the vellum paper to write religious texts. In 1906 J.L. Heiberg translated the text, decoding large chunks of the erased information. He found two calculus methods that exactly summed the area of a parabola by infinite and finite slices. The calculus methods may not have primarily used the method of exhaustion, as often reported in math history texts. The infinite slice method, linked to the proposed 'method of exhaustion' method is clearly mentioned. \PMlinkexternal{E.J. Dijksterhuis}{http://mathforum.org/kb/message.jspa?messageID=5847373&tstart=90} included Heiberg's view of the information in a 1987 biography of Archimedes, published by Princeton Press, begins with Archimedes' Lemma per paragraph "7.60 Q.P.23 In Quadrature of the Parabola Archimedes proves the following proposition on the sum of a geometrical progression with a common ratio of 1/4. Given a series of magnitudes, each of which is equal tofour times the order of the next, all of the magnitudes and one-third of the least added together will exceed the greatest by one-third. Let the magnitudes A, B, C, D, E be given such that it has to be proven that A + B + C + D + E + 1/3E = (4/3)A Dijksterhuis wrote out the 1/4 geometric infinite series: $$4A/3 = A + A/4 + A/16 + A/64 + ... $$ This is the infinite series method. Heiberg published the finite Egyptian fraction series: $$4A/3 = A + A/4 + A/12$$ in a way that proved or completed the 1/4 geometric series problem. The palimpsest document came on the open market a few years ago. It was auctioned for 2,000,000 dollars. NOVA reported a revised analysis of the text that was suggested by its new owners. The NOVA program did not include Heiberg and Dijksterhuis' 1/4 geometric series method written as a finite series in its review. \PMlinkexternal{Stanford University investigators}{http://www.archimedespalimpsest.org/mediacenter_presskit.html} only published the infinite series side of the document, outlined the above, without showing the simple aspects of the 1/4 geometric finite series, $$1/3 = 1/4 + 1/16 + 1/64 + ... + 1/4n + ... $$ which is one-half phase of the Horus-Eye series: $$1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/2n + ... $$ which Archimedes may have extended Eudoxian (and Egyptian) traditions that generally converted several classes of binary and closely related infinite series to exact finite Egyptian fraction series proofs.