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'Archimedes' calculus'
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| Title of object: |
Archimedes' calculus |
| Canonical Name: |
ArchimedesCalculus |
| Type: |
Definition |
| Created on: |
2008-07-14 09:44:18 |
| Modified on: |
2008-07-21 08:53:47 |
| Classification: |
msc:01A20 |
| Synonyms: |
Archimedes' calculus=differential calculus |
Preamble:
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Content:
The differential calculus of Archimedes has been passed down to the modern era within one text, an erasable parchment (palimpsest). The information had not been 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 read the text, decoding large chunks of the erased information. He found a calculus method that exactly sums an infinite slice of a parabola. The calculus method did not use the method of exhaustion as often reported in math history texts. \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:
$$4A/3 = A + A/4 + A/16 + A/64 + ... $$
proven by a finite Egyptian fraction series:
$$4A/3 = A + A/4 + A/12$$
which completes the problem.
As many recall, the 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} have published one translation of the document, outlining the above, without showing the simple aspects of the 1/4 geometric 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 extended older Eudoxian (and Egyptian) traditions that generally converted several classes of binary and closely related infinite series to finite Egyptian fraction series proofs.
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