Archimedes' calculus

Archimedes is given credit for publishing the first known calculus. Fragmented records show that Archimedes recorded two methods the defined aspects of the earliest calculus. Which method was primary offers a logical story that is not easy to follow.

The traditional story reported in math history texts says that Archimedes "method of exhaustion " that defined the area of a parabola on an erasable parchment (palimpsest). The parchment's numerical information was not recorded in Archimedes' handwriting. Worse, the parchment's information was copied over hundreds of years, and erased in 1,100 AD by Byzantine priests. Byzantines used the vellum parchment to write religious texts.

The first calculus recorded a possible second second that followed the first method. In 1906 J.L. Heiberg translated portions of the hard-to-read text. Heiberg showed that the first method exactly summed the area of a parabola by an infinite 1/4 geometric series, the method of exhaustion, as a statement of the problem.

Archimedes may have asked. "How can an infinite series be exactly summed to a finite series"?

The second method wrote out a finite Egyptian fraction series, exactly pointed out the answer.

Considering both methods, Archimedes' calculus stated a 1/4 geometric series (algorithm) problem solved by a finite unit fraction series solution.

Aspects of the longer story were recently reported by Stanford University researchers. Researchers stressed the infinite series algorithm side of the document as Archimedes' solution. The finite Egyptian fraction information published by Heiberg in 1906 and Dijksterhuis in 1987 were ignored by Stanford researchers as a possible primary solution.

E.J. Dijksterhuis included Heiberg's view in the 1987 "Archimedes" biography published by Princeton Press, The discussion begins with an Archimedes Lemma: 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 to four 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

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 + ...

an infinite series.

Heiberg published a finite Egyptian fraction series side of the discussion, as Dijksterhuis wrote as:

4A/3 = A + A/4 + A/12

that proved the accuracy of a finite 1/4 geometric series method that followed Eudoxus that used the same tradition.

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. Stanford University investigators only published the infinite series (1/4 geometric series) side of the document without discussing the equally important Egyptian fraction series side connecting:

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 + ...

CONCLUSION: Archimedes is proposed as creating the first calculus by first stating the problem (finding the area of parabola) as as infinite series, and second applying a finite arithmetic method. Taking both methods as one method, Archimedes may have converted rational numbers by a 1/4 geometric infinite series, and proved the accuracy of the infinite series by calculating a finite series.