Ivars Peterson's MathTrek
"What we are told about Archimedes is a mix of a few hard facts and many legends," Sherman Stein of the University of California, Davis notes in his book Archimedes: What Did He Do Besides Cry Eureka?
I was reminded of that statement when my son Kenneth and I recently visited a bookstore and happened to hear the children's storyteller Jim Weiss. In a highly entertaining, evocative manner, he narrated the famous tale of how Archimedes of Syracuse (287–212 BC) used the displacement of water to uncover a scam, demonstrating that a wreath-shaped crown actually consisted of a mixture of silver and gold rather than pure gold.
The first known appearance of the story of this feat is in the writings of the Roman architect and engineer Vitruvius, who lived in the first century BC.
"While Archimedes was turning the problem over, he chanced to come to the place of bathing, and there, as he was sitting down in the tub, he noticed that the amount of water which flowed over the tub was equal to the amount by which his body was immersed. This showed him a means of solving the problem, and he did not delay, but in his joy leapt out of the tub and, rushing naked towards his home, he cried out with a loud voice that he had found what he sought," Vitruvius recounted in the book De Architectura.
As told by Vitruvius, the tale has some implausible features. It seems unlikely, for example, that Archimedes had measuring equipment of sufficient accuracy to detect the rather small difference in displacement between a crown made of pure gold and one fashioned from alloyed gold. See http://www.mcs.drexel.edu/~crorres/Archimedes/Crown/CrownIntro.html for additional details.
Elsewhere in his book, Vitruvius gives a detailed description of the Archimedes screw, an ingenious device used to lift water for irrigation or drainage. Although this ancient machine is often attributed to Archimedes, it isn't clear whether he was its inventor. On the other hand, there is no known mention of the device before his time.
Mathematician Chris Rorres of Drexel University in Philadelphia has taken a close look at Vitruvius's specifications for constructing an Archimedes screw.
Construction of the screw begins with a tree trunk shaped into a cylindrical core, whose length is 16 times its diameter. Eight intertwined helical blades are then attached to the core, each blade having a width equal to the core's radius. The period, or "pitch," of these blades equals the cylindrical core's circumference. An outer covering of wooden planks encloses the mechanism. The screw is then mounted so that it is tilted in the direction of the hypotenuse of a 3-4-5 right-angled triangle. The bottom end is immersed in a basin of water. Rotating the screw raises water from the basin to an upper reservoir.
"I have now described as clearly as I could, to make them better known, the principles on which wooden engines for raising water are constructed, and how they get their motion so that they may be of unlimited usefulness through their revolutions," Vitruvius concluded.
How good was Vitruvius's design? Rorres worked out the geometry that would maximize the amount of water delivered from a lower basin to an upper reservoir in one turn of the screw for a given outer radius of the screw.
A screw's location and how much water is to be lifted usually determine its outer radius, length, and slope, Rorres says. Its inner radius, the number of blades, and the pitch of the blades can then be chosen to optimize the screw's performance.
In the January ASCE Journal of Hydraulic Engineering, Rorres reports that the output of Vitruvius's screw is fairly close to that of the optimal 8-bladed screw. In addition, its ratio of inner to outer radius is within 7 percent of the optimal value, and its pitch ratio is within 27 percent.
At the same time, "the construction lines associated with its design are much simpler than those that would be needed to construct the optimal screw," Rorres notes. "No doubt, many generations of experience went into the design of the screw that Vitruvius described."
Recent years have seen a revival of interest in the Archimedes screw, particularly for its proven, trouble-free design and its ability to lift debris-laden water effectively, Rorres remarks. "It has also proved valuable in installations where damage to aquatic life must be minimized."
Rorres, C. 2000. The turn of the screw: Optimal design of an Archimedes screw. ASCE Journal of Hydraulic Engineering 126(January):72-80. Article available at http://www.mcs.drexel.edu/~crorres/screw/screw.pdf. Other information can be found at http://www.mcs.drexel.edu/~crorres/Archimedes/Screw/optimal/optimal.html.
Stein, S. 1999. Archimedes: What Did He Do Besides Cry Eureka? Washington, D.C.: Mathematical Association of America. (See http://www.maa.org/pubs/books/arc.html.)
Information about Archimedes and his discoveries and inventions can be found at http://www.mcs.drexel.edu/~crorres/Archimedes/contents.html.
Jim Weiss tells the story of Archimedes and the golden crown on the audio CD Galileo and the Stargazers, produced by Greathall in Charlottesville, Va. (http://www.greathall.com/onlinebro.html).
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