The Archimedes Codex: A Prayer & Origins of Calculus

For decades, a prayer book sat in the closet of a family in France. The owners of the book noticed that beneath the prayer text seemed to be faint Greek letters with an occasional diagram cut off by the spine. Wondering if the old book had some value, they brought it to Christie's Auction House of London where it was auctioned off for two million dollars.

Turns out it wasn’t just a prayer book, but the only surviving copies of several works by the ancient mathematician Archimedes.

The surface layer of writing describes Christian prayers, but underneath were found the only surviving copies of many works of the ancient Greek mathematical genius Archimedes.

Intensive research over the last nine years has revealed an astounding discovery. Archimedes was working out principles that would form the basis of calculus. It was also revealed that he had a very sophisticated mathematical concept of infinity, much more so than anyone had realized.

Archimedes wrote his manuscript on a papyrus scroll 2,200 years ago. Later, someone copied the text from papyrus to animal-skin parchment. Then, just 700 years ago, a monk decided to make use of the old parchment for a new prayer book. He took the old copy of Archimedes' book, cut the pages in half, rotated them 90 degrees, and scraped the surface to remove the ink. This was a fairly common practice for making oneself some “fresh” writing material, especially during a time and place where religion was seen as more important than science. The monk was then able to write his prayers on the nearly blank pages.

But fortunately, the monk didn’t do a perfect job of wiping away Archimedes works. The book's anonymous buyer funded an extensive project to stabilize the condition of the book. Next researchers took digital pictures of it in different wavelengths of light, creating a multi-spectral image that could be manipulated to reveal the text by Archimedes. On four of the pages, forged paintings covered the entire text, so the researchers used x-ray fluorescence imaging to peek beneath the paintings and decipher the obscured text. One of Archimedes's works discovered was simply titled The Method. Turns out, it is the earliest known work on calculus.

Archimedes wrote The Method almost two thousand years before Isaac Newton and Gottfried Wilhelm von Leibniz developed calculus in the 1700s. Reviel Netz, an historian of mathematics at Stanford University who transcribed the text, says that the examination of Archimedes' work has revealed "a new twist on the entire trajectory of Western mathematics."

In The Method, Archimedes was working out a way to compute the areas and volumes of objects with curved surfaces, which was also one of the problems that motivated Newton and Leibniz. Ancient mathematicians had long struggled to "square the circle" by calculating its exact area. That problem turned out to be impossible using only a straightedge and compass, the only tools the ancient Greeks allowed themselves. Nevertheless, Archimedes worked out ways of computing the areas of many curved regions.

The Greek philosopher Aristotle built defenses against infinity's paradoxical qualities by distinguishing between the "potential infinite" and the "actual infinite." An infinitely long line would be actually infinite, whereas a line that could always be extended would be potentially infinite. Aristotle argued that the actual infinite didn't exist.

Archimedes developed rigorous methods of dealing with infinity, which are still used today. Archimedes proved that the area of a section of a parabola is four-thirds the area of the triangle inside it. To do so, he built a straight-lined figure that's an approximation of the curvy one. Then he showed that he could make the approximation as close as anyone could ever demand to both the section of the parabola and to four-thirds the area of the triangle.

Netz and the project's lead researcher, William Noel of the Walters Art Museum in Baltimore, have co-authored a new book, The Archimedes Codex: Revealing The Secrets Of The World's Greatest Palimpsest about the discovery, which is scheduled for release on Nov. 1 of this year.

"The interesting breakthrough is that he is completely willing to operate with actual infinity," Netz says, but he adds that "the argument is definitely not completely valid. He just had a strong intuition that it should work." In this case, it did work, but it remained for Newton and Leibniz to figure out how to make the argument mathematically rigorous.

Newton and Leibniz also worked with actual infinity. Leibniz went so far as to say in a letter, "I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author."

Modern calculus no longer makes use of the actual infinite; it sticks with Aristotle's distinction. Philosophers still argue over the legitimacy of the notion of actual infinity. Netz argues, however, that The Method reveals the originality and daring of Archimedes's thought and shows that he anticipated some of the bold steps that would later lead to the full development of calculus.

Posted by Rebecca Sato

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links:
http://www.sciencenews.org/articles/20071006/mathtrek.asp
http://archimedespalimpsest.org/mediacenter_articles-4-26-07.html

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Archimedean Palimpsest
by Christine Crockett

Pressed from a quill, the umber bleeds
into goat-hide, traces his spiraling quest
for bodies afloat in quantified bliss (oh,
the equilibrium of parallel planes!)

until monks in the silence of limestone cells,
pumiced down proofs, glazed them with grace.
(How could the mechanics of displaced weight
trump a trinity fixed as a ring of stars?)

Yet, determined, new arcs unravel such faith;
impatiently, doubts await their proofs:
milk-white electrons, trapped then flung
from a curve quickened to speed of light,

illuminate logic asleep beneath skin.
The sole source known to irradiate
iron, burns past fingerprints, grease
and bone, past umber mortared to finest

dust, past God's holy saints into space-spun
time, where all particles slow to a murky
stillness, and delicate fibers surrender joys
that his soul merely stored and remembered.

Posted by: christine crockett | October 10, 2007 at 05:06 PM