Zero - Charles Seife [42]
In one of Kepler’s lesser-known works, Volume-Measurement of Barrels, he does this in three dimensions, slicing barrels into planes and summing the planes together. Kepler, at least, wasn’t afraid of a glaring problem: as ?x goes to zero, the sum becomes equivalent to adding an infinite number of zeros together—a result that makes no sense. Kepler ignored the problem; though adding infinite zeros together was gibberish from a logical point of view, the answer it yielded was the right one.
Kepler was not the only prominent scientist who sliced objects infinitely thin. Galileo, too, pondered infinity and these infinitely small slices of area. These two ideas transcend our finite understanding, he wrote, “the former on account of their magnitude, the latter because of their smallness.” Yet despite the deep mystery of the infinite zeros, Galileo sensed their power. “Imagine what they are when combined,” he wondered. Galileo’s student Bonaventura Cavalieri would provide part of the answer.
Instead of barrels, Cavalieri cut up geometric objects. To Cavalieri, every area, like that of the triangle, is made up of an infinite number of zero-width lines, and a volume is made up of an infinite number of zero-height planes. These indivisible lines and planes are like atoms of area and volume; they can’t be divided any further. Just as Kepler measured the volumes of barrels with his thin slices, Cavalieri added up an infinite number of these indivisible zeros to figure out the area or the volume of a geometric object.
For geometers, Cavalieri’s statement was troublesome indeed; adding infinite zero-area lines could not yield a two-dimensional triangle, nor could infinite zero-volume planes add up to a three-dimensional structure. It was the same problem: infinite zeros make no logical sense. However, Cavalieri’s method always gave the right answer. Mathematicians ignored the logical and philosophical troubles with adding infinite zeros—especially since indivisibles or infinitesimals, as they came to be called, finally solved a long-standing puzzle: the problem of the tangent.
A tangent is a line that just kisses a curve. For any point along a smooth curve that flows through space, there is a line that just grazes the curve, touching at exactly one point. This is the tangent, and mathematicians realized that it is extremely important in studying motion. For instance, imagine swinging a ball on a string around your head. It’s traveling in a circle. However, if you suddenly cut the string, the ball will fly off along that tangent line; in the same way, a baseball pitcher’s arm travels in an arc as he throws, but as soon as he lets go, the ball flies off on the tangent (Figure 24). As another example, if you want to find out where a ball will come to rest at the bottom of a hill, you look for a point where the tangent line is horizontal. The steepness of the tangent line—its slope—has some important properties in physics: for instance, if you’ve got a curve that represents the position of, say, a bicycle, then the slope of the tangent line to that curve at any given point tells you how fast that bicycle is going when it reaches that spot.
Figure 24: Flying off at a tangent
For this reason, several seventeenth-century mathematicians—like Evangelista Torricelli, René Descartes, the Frenchman Pierre de Fermat (famous for his last theorem), and the Englishman Isaac Barrow—created different methods for calculating the tangent line to any given point on a curve. However, like Cavalieri, all of them came up against the infinitesimal.
To draw a tangent line at any given point, it’s best to make a guess. Choose another point nearby and connect the two. The line you get isn’t exactly the tangent line, but if the curve isn’t too bumpy, the two lines will be pretty close. As you reduce the distance between the points, the guess gets closer to the tangent line (Figure 25). When your points are zero distance away from each other, your approximation is perfect: you have found the tangent. Of course, there’s a problem.
Figure 25: Approximating