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Archimedes and Euclid? Like String Theory versus Freshman Calculus


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archimedes spiral

Archimedes spiral imaged in UV rays, from the Archimedes Palimpsest, which will be showcased next month at the Walters Art Museum in Baltimore

The archetype of the science genius didn’t use to be Albert Einstein. For centuries, the quintessential irreverent, visionary scientist, immersed in a world of his own making to the point of forgetting to put on his clothes, was instead an ancient Greek mathematician. His name was Archimedes of Syracuse.

Archimedes was the stuff of legend around the Mediterranean world—what with the jumping out of his bath tub, the mirrors he used to set Roman ships on fire, the lever that he said he could have used to lift the world—even while he was alive. But the “Eureka!” incident may be a good indicator of the extent to which the lore failed to capture the true significance of his mind.

The story says that Archimedes had an epiphany while taking a bath, jumped out of the water and ran naked down the street screaming “Eureka!” which means “I’ve found it!” Supposedly, while contemplating the floating of his own body in the tub, Archimedes had realized that he could solve a riddle proposed by King Hiero II.

The king wanted to know whether his crown was made of pure gold, as stipulated, or whether the goldsmith had mixed some silver in. Archimedes’ answer: dip the crown in a tub and see how much water it displaces. That way you can measure the crown’s volume, and from that you can then calculate its density and thus guess at its composition.

But I think it is unlikely that a mind of Archimedes’ caliber would have gotten all worked up about such a triviality. If there is anything true to the legend, perhaps it’s that while in his bath tub Archimedes figured the law of buoyancy: the principle that still bears his name. Far from being a sideshow trick of Monty Pythonesque flavor, it was a true scientific law, one that applies whether it’s a scientist in his bath tub or a hot air balloon floating in the air.

In the October Scientific American I write about an exhibition that will open next month at the Walters Art Museum in Baltimore, showcasing the incredible vicissitudes of one of just three medieval copies of Archimedes’ works that  survived through the Dark Ages “by the narrowest of threads,” as the manuscript’s curator, Will Noel, puts it. The book is called the Archimedes Palimpsest.

archimedes palimpsest

This 10th-century prayer book was made of recycled parchment. Before being washed out and reused, the pages contained the only surviving copies of two of Archimedes' works. It is now known as the Archimedes Palimpsest.

 

The story of the Palimpsest, as Noel said to me in an interview several years ago, is the “Jurassic Park of manuscripts.” It is a story that has been told on many occasions, nowhere more poignantly than in The Archimedes Codex, a book Noel co-authored in 2007 with Reviel Netz, a historian of mathematics at Stanford University. Theirs is a must-read book for anyone who has an interest in archeology or ancient history or ancient mathematics–or simply for anyone who likes a damn good story.

Noel assembled a team of some of the world’s best imaging experts to recover as much as possible of Archimedes’ text from the Palimpsest that no eyes had seen in modern times. One of them was physicist Uwe Bergmann at the Stanford Synchrotron Radiation Lab, who starting in 2006 scanned some of the pages with x-rays from a particle accelerator: during the scans, Noel watched ecstatically as the images slowly appeared on the computer screen, pixel by pixel, which was “like receiving a fax from the 3rd Century B.C.,” Noel told the BBC at the time.

x-rayed text

Detail of an x-ray scan of a page of the Archimedes Palimpsest, showing the prayer text (vertical) and the older text of a 10th-century copy of Archimedes (horizontal), in red. The x-rays revealed iron contained in the ink that had seeped into the parchment.

One of the incredible facts I learned from Noel and Netz’s book is that to this day, few of Archimedes’ works have even been translated into English. T. L. Heath’s 1897 edition of Archimedes’ works—at least the ones available to him back then, before the Palimpsest was rediscovered—was admittedly more a paraphrase than a translation.

Netz made it his lifelong mission to transcribe and translate the 100,000-odd words of text that survive from Archimedes’s writings, including, for the first time, a critical edition of the figures, which older editions usually tended to leave aside or to interpret in their own way.

In fact, although Archimedes always had the reputation of being the greatest mathematician of the ancient world, very few people ever read much of his works because few could understand them, says Netz.

For two millennia Euclid’s Elements had its place as a geometry textbook and a paragon of rational thought. While Euclid (who may have been one of Archimedes’ mentors when the Syracusan spent time in Alexandria) was a great systematizer and a master of exposition, Archimedes wrote about his original findings, at too high a level for most people, Netz says.

As Netz writes in one of the chapters he authored in the Archimedes Codex (he and Noel took turns, each writing every other chapter), Archimedes did not hold Euclid’s Elements in much esteem. “Archimedes wouldn’t think very highly of them, as they consist mainly of basic mathematics. Archimedes was an advanced mathematician, writing for people who knew much more than the contents of Euclid’s Elements.”

Compared to reading Euclid, reading Archimedes may have been a bit like reading an abstruse string theory article versus reading a college physics textbook, or perhaps one of calculus for freshmen.

Most of Archimedes’ writings were in the form of letters written to people whose minds Archimedes considered worthy of his attention. It was in one such letter that Archimedes made use of the concept of infinity, according to Noel. That letter was addressed to Eratosthenes, one of the most eminent intellectuals of the time—the man who, with incredible ingenuity, had calculated the radius of the Earth to within a few percent points of its actual value (a story masterfully told by Carl Sagan in the first episode of Cosmos).

Among those few who later understood Archimedes were Galileo Galilei, Isaac Newton and Gottfried Wilhelm von Leibniz, and it is not a coincidence that these men were the founders of modern science, Netz says. In particular, Archimedes’ use of infinitesimals to calculate areas and volumes was the starting point for the invention of the calculus, he says. And the modern mathematicians explicitly credited Archimedes for their use of infinitesimals, says Netz. “It was not a rediscovery by any means.”

Further readings:

Images courtesy of the anonymous owner of the Archimedes Palimpsest and the Walters Art Museum

About the Author: Davide Castelvecchi is a freelance science writer based in Rome and a contributing editor for Scientific American magazine. Follow on Twitter @dcastelvecchi.

The views expressed are those of the author and are not necessarily those of Scientific American.






Comments 5 Comments

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  1. 1. pulpus 2:48 pm 09/20/2011

    Already read that book a few months ago. Amazing how near to the basics of calculus Archimedes was!!
    Do you have some news about the availability of the transcritpion?

    Link to this
  2. 2. @dcastelvecchi 3:02 pm 09/20/2011

    The transcriptions are due to be published later this year (October 31 according to Amazon). For the translation it will take a lot longer … can you read Doric Greek? :-)

    http://www.amazon.com/Archimedes-Palimpsest-Set-Publications/dp/1107016843/ref=sr_1_1?s=books&ie=UTF8&qid=1316545227&sr=1-1

    Link to this
  3. 3. SAJP2000 6:50 pm 09/20/2011

    What good fortune to have discovered this work–a work of incredible genius erased and written over by god-worshipers.

    It’s somewhat painful to consider how many other similarly profound works have been erased, burned/destroyed such as those at Alexandria or remain hidden away in the Vatican Library by religious zealots.

    What a truly wonderful effort by Noel and Netz. Bravo gentlemen!

    Link to this
  4. 4. fcfcfc 5:51 am 03/24/2012

    http://www.guccisale2012.net

    Link to this
  5. 5. sharpnova 8:57 pm 08/14/2013

    Yeah. No. I’ve read a ton of both Archimedes and Euclid’s works.

    Euclid’s was the greater mind.

    Link to this

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