The Calculus Diaries - Jennifer Ouellette [7]
To fully appreciate the significance of this discovery, one needs a bit of historical context. It all started with the problem of curves. In the beginning, there was Euclid, whose geometrically simple world consisted of tidy planes, clean straight lines, and points, with an arc or a circle tossed in now and then, just to add a bit of variety to the mix. It was all lovingly laid out in a thirteen-volume treatise called the Elements, perhaps the most influential mathematical text of all time.5 Conspicuously lacking in Euclid’s collection of geometric postulates and axioms were curves. He was certainly familiar with curvy shapes. There is evidence in the writings of a later Greek mathematician, Apollonius of Perga (circa 160 B.C.) that Euclid studied cross sections of cones, although any treatise he may have written on the subject has not survived.
Cones slice and dice into four basic curves: the circle, the ellipse, the parabola, and the hyperbola. What curve you get depends on the angle of the plane as you’re slicing. A simple horizontal slice, for instance, will yield a circle. Tip the plane very slightly, and that circle becomes an ellipse. Tilt the plane such that it runs parallel to one side of the cone, and that ellipse becomes a parabola. Finally, if you tilt the plane so that it intersects the second cone, you end up with a hyperbolic curve.
There are many other kinds of curves, of varying complexity, but they are much more than mere geometric curiosities. Curves are geometry in motion. A pendulum swinging back and forth forms the arc of a circle. A falling apple forms a parabola, as does the trajectory of a baseball. The planets move about the sun in elliptical orbits, while many comets move in hyperbolic orbits. A spring bouncing in the absence of friction forms a periodic sine wave. All these types of motion can be represented graphically by a smooth, continuous curve, and that curve in turn can be used to make predictions about the trajectory (path) of a moving object. That makes geometric curves central to the realm of calculus, so naturally that’s where I started in my mathematical quest.
DANGEROUS CURVES
Archimedes liked to invent things, and he had a particular knack for fashioning ingenious weapons of war. Legend has it that he built an array of gigantic mirrors of bronze or glass and arranged them in such a way that they were capable of collecting, focusing, and redirecting the sun’s rays in order to set enemy ships on fire from a distance—a scaled-up version of frying bugs by focusing sunlight onto them with a magnifying glass. Ancient historical accounts report that he turned this ingenious “death ray” device onto invading Roman ships during the siege of Syracuse in 213 B.C., reducing them to cinders.6
The best shape for an array of mirrors in order to accomplish this is a parabola. Being a thorough sort of inventor, Archimedes set about figuring out how to determine the area under that particular curve. Curves were knotty problems in Euclidean geometry. It’s a simple enough matter to determine the area of a triangle or a rectangle, with their clean, straight lines, but ancient mathematicians wrestled mightily with a means for determining the area under a curve. One hundred years or so before Archimedes, a Greek astronomer and mathematician named Eudoxus of Cnidus figured out that while one couldn’t calculate the area under a curve exactly, it was possible to approximate that area by filling it in with a succession of rectangles (see above).
Now you just need to figure out the area for each of those rectangles by multiplying the width by the height. Then add them up, and the result is a rough estimate of the area under the curve. That is known as a first approximation.