The Calculus Diaries - Jennifer Ouellette [8]
Archimedes found himself adapting Eudoxus’s method to design his ship-incinerating death ray. He knew quite well how to determine the area of a triangle. So Archimedes drew a triangle inside the parabola, leaving two small gaps. He then drew two smaller triangles inside those gaps, leaving four even smaller gaps, and continued to draw ever-smaller triangles to fill the ever-smaller gaps. Then he calculated the area of each triangle and added them together to get an approximation of the area under the parabola.
Even though the end result was an approximation, it was close enough to the precise area to suit the Greek inventor’s needs. More important, it was a critical first step toward defining calculus. With each iteration in the method of exhaustion, the triangles become smaller and smaller, and thus it takes more and more of them to fill the area under the curve. When the number of triangles (or rectangles, in Eudoxus’s original method) becomes infinite, that is the point where we get the exact area under the curve. And that process of summing up an infinite series of things is the essence of integral calculus.
Archimedes actually may have been less successful at building a viable death ray than he was at estimating the area under a curve. Numerous attempts have been made over the years to re-create this supposedly pivotal moment in the siege of Syracuse, most recently in 2005 by a team of engineering students at MIT. The team built an enormous bronze and glass reflector on the edge of San Francisco bay and tried to focus sunlight onto a small fishing boat about 150 feet away, in hopes of setting it on fire. This didn’t work. So the MIT engineers moved the boat closer, to around 75 feet. This time they managed to create a small fire, although it quickly fizzled out.
Part of the problem was cloud cover; the mirrors only work when the sun is shining. Since Syracuse faced east toward the ocean, Archimedes’ device would have only been useful in the morning. Then there is the time factor: the death ray did not work quickly. Shooting flaming arrows at the Roman ships anchored in the harbor would have been far more practical and efficient. That was the conclusion of TV’s Mythbusters, who issued the challenge to MIT after failing in their own attempt to re-create the boat-burning. Executive producer Peter Rees told the Guardian that the tale of Archimedes’ death ray is mostly likely a myth: “We’re not saying it can’t be done. We’re just saying it’s extremely impractical as a weapon of war.”
Even if the weapon proved impractical, the exercise of creating it gave Archimedes some valuable insights into geometric curves. His array of flat mirrors formed a makeshift parabola out of straight lines; together they approximated a parabolic curve. Magnify any curved line sufficiently, and it looks more and more like a straight line with each level of magnification. Archimedes realized he could view a circle, for example, dynamically as an accumulation of an infinite number of smaller pieces added together—triangles, again, in this case—rather than as a static, unchanging whole.
This is the method he used to prove how to find the area of a circle: half the product of its radius and its circumference. It’s now a standard maxim in geometry textbooks. It worked out so well that Archimedes later adapted Eudoxus’s method to calculate the volume of a sphere