Science News / A Prayer For Archimedes

For seventy years, a prayer book moldered in the closet of a family in France, passed down from one generation to the next. Its mildewed parchment pages were stiff and contorted, tarnished by burn marks and waxy smudges. Behind the text of the prayers, faint Greek letters marched in lines up the page, with an occasional diagram disappearing into the spine.

Image: The top layer of writing in this 700-year-old book describes Christian prayers. But underneath, almost obliterated, are the only surviving copies of many of the works of the ancient Greek mathematician Archimedes.

The owners wondered if the strange book might have some value, so they took it to Christie's Auction House of London. And in 1998, Christie's auctioned it off—for two million dollars.

For this was not just a prayer book. The faint Greek inscriptions and accompanying diagrams were, in fact, the only surviving copies of several works by the great Greek mathematician Archimedes.

An intensive research effort over the last nine years has led to the decoding of much of the almost-obliterated Greek text. The results were more revolutionary than anyone had expected. The researchers have discovered that Archimedes was working out principles that, centuries later, would form the heart of calculus and that he had a more sophisticated understanding of the concept of infinity than anyone had realized.

Image: Archimedes computed the area of the curved figure (left) by enclosing it in a bigger one with straight edges (right). He then examined random slices to compute the volume—using the concept of actual infinity.

Archimedes wrote his manuscript on a papyrus scroll 2,200 years ago. At an unknown later time, someone copied the text from papyrus to animal-skin parchment. Then, 700 years ago, a monk needed parchment for a new prayer book. He pulled the copy of Archimedes' book off the shelf, cut the pages in half, rotated them 90 degrees, and scraped the surface to remove the ink, creating a palimpsest—fresh writing material made by clearing away older text. Then he wrote his prayers on the nearly-clean pages.

... Archimedes wrote The Method almost two thousand years before Isaac Newton and Gottfried Wilhelm von Leibniz developed calculus in the 1700s. ... In The Method, Archimedes was working out a way to compute the areas and volumes of objects with curved surfaces...

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 other curved regions.

Such problems are tricky because solving them directly requires slicing up curved areas into infinitely many areas with straight boundaries. But the concept of infinity is a slippery and troublesome one that can quickly lead to paradox.

The Greek philosopher Aristotle built defenses against infinity's vexing 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.

... Modern historians and mathematicians have always believed whenever Archimedes dealt with infinities, he kept strictly to the potential kind. But Netz, who transcribed the newly found text, says that the recent discoveries show that Archimedes indeed used the notion of actual infinity.

Archimedes's key argument about infinity appears on pages so damaged that Heiberg had been unable to transcribe them. Archimedes calculated the volume of a body shaped something like a fingernail by enclosing it in a volume bounded by plane surfaces. But instead of making better and better approximations of the curved figure, as he had done with the parabolic section, he pondered a two-dimensional slice through the larger volume enclosing the smaller one.

Archimedes found a relationship between the full area of that slice, which was a section through the plane-sided volume, and the smaller area within it, which was a section through the curved shape. Then he argued that he could use that relationship to calculate the entire volume of the curved shape, because both the curved figure and the straight one contained the same number of slices. That number just happened to be infinity—actual infinity.

"The interesting breakthrough is that he is completely willing to operate with actual infinity," Netz says....
via Science News / A Prayer For Archimedes.

Reading about the great accomplishments of people like Archimedes and Einstein makes me want to add some significant contribution to science. Finding a cheap clean alternative energy source as convenient and energy dense as gasoline, reversing gravity, curing aging, something like that...