Zero - Charles Seife [44]
Multiplying out the (x + o)2 term gives us
Rearranging the terms yields
Since y = x2 + x + 1, we can subtract y from the left side of the equation and x2 + x + 1 from the right side of the equation and leave the system unchanged. That leaves us with
Now comes the dirty trick. Newton declared that since o was really, really small, (o)2 was even smaller: it vanished. In essence, it was zero, and could be ignored. That gives us
which means that o/o = 2x + 1, which is the slope of the tangent line at any point x on the curve (Figure 26). The infinitesimal time period o drops right out of the equation, o/o becomes /, and o need never be thought of again.
The method gave the right answer, but Newton’s vanishing act was very troubling. If, as Newton insisted, (o)2 and (o)3 and higher powers of o were equal to zero, then o itself must be equal to zero.* On the other hand, if o was zero, then dividing by o as we do toward the end is the same thing as dividing by zero—as is the very last step of getting rid of the o in the top and bottom of the o/o expression. Division by zero is forbidden by the logic of mathematics.
Figure 26: To find the slope at a point on the parabola y=x2 + x + 1, use the formula 2x + 1.
Newton’s method of fluxions was very dubious. It relied upon an illegal mathematical operation, but it had one huge advantage. It worked. The method of fluxions not only solved the tangent problem, it also solved the area problem. Finding the area under a curve (or a line, which is a type of curve)—an operation we now call integration—is nothing more than the reverse of differentiation. Just as differentiating the curve y = x2 + x + 1 gives you an equation for the slope of the tangent—y = 2x + 1—integrating the curve y = 2x + 1 gives you a formula for the area under the curve. This formula is y = x2 + x + 1; the area underneath the curve between the boundaries x = a and x = b is simply (b2 + b + 1) - (a2 + a + 1) (Figure 27). (Technically, the formula is y = x2 + x + c, where c is any constant you choose. The process of differentiation destroys information, so the process of integration doesn’t give you exactly the answer you are looking for unless you add another bit of information.)
Calculus is the combination of these two tools, differentiation and integration, in one package. Though Newton broke some very important mathematical rules by toying with the powers of zero and infinity, calculus was so powerful that no mathematician could reject it.
Nature speaks in equations. It is an odd coincidence. The rules of mathematics were built around counting sheep and surveying property, yet these very rules govern the way the universe works. Natural laws are described with equations, and equations, in a sense, are simply tools where you plug in numbers and get another number out. The ancients knew a few of these equation-laws, like the law of the lever, but with the beginning of the scientific revolution these equation-laws sprang up everywhere. Kepler’s third law described the time it takes for planets to complete an orbit: r3/t2 = k for time t, distance r, and a constant k. In 1662, Robert Boyle showed that if you take a sealed container with a gas in it, squishing the container would increase the pressure inside: pressure p times volume v was always a constant—pv = k for a constant k. In 1676, Robert Hooke figured out that the force exerted by a spring, f, was a negative constant, –k, multiplied by the distance, x, that you’ve stretched it: f =-kx. These early equation-laws were extremely good at expressing simple relationships, but equations have limitations—their constancy, which prevented them from being universal laws.
Figure 27: To find the area under the line