Zero - Charles Seife [45]
As an example, let’s take the famous equation we all learned in high school: rate times time equals distance. It shows how far you get, x miles, when you run with a certain velocity, v miles per hour, for a time, t hours: vt = x; after all, miles per hour times hours equals miles. This equation is very useful when you are calculating how long it will take to get from New York to Chicago on a train that moves exactly 120 miles an hour. But how many things really move at a constant rate like a train in a math problem? Drop a ball, and it moves faster and faster; in this case, x = vt is quite simply wrong. For the case of a dropped ball, x = gt2/2, where g is the acceleration due to gravity. On the other hand, if you’ve got an increasing force on the ball, x might equal something like t3/3. Rate times time equals distance is not a universal law; it doesn’t apply under all conditions.
Calculus allowed Newton to combine all these equations into one grand set of laws—laws that applied in all cases, under all conditions. For the first time, science could see the universal laws that underlie all of these little half laws. Even though mathematicians knew that calculus was deeply flawed—thanks to the mathematics of zero and infinity—they quickly embraced the new mathematical tools. For the truth is, nature doesn’t speak in ordinary equations. It speaks in differential equations, and calculus is the tool that you need to pose and solve these differential equations.
Differential equations are not like the everyday equations that we are all familiar with. An everyday equation is like a machine; you feed numbers into the machine and out pops another number. A differential equation is also like a machine, but this time you feed equations into the machine and out pop new equations. Plug in an equation that describes the conditions of the problem (is the ball moving at a constant rate, or is a force acting on the ball?) and out pops the equation that encodes the answer that you seek (the ball moves in a straight line or in a parabola). One differential equation governs all of the uncountable numbers of equation-laws. And unlike the little equation-laws that sometimes hold and sometimes don’t, the differential equation is always true. It is a universal law. It is a glimpse at the machinery of nature.
Newton’s calculus—his method of fluxions—did just this by tying together concepts like position, velocity, and acceleration. When Newton denoted position with the variable x, he realized that velocity is simply the fluxion—what modern mathematicians call the derivative—of x: And acceleration is nothing more than the derivative of velocity, Going from position to velocity to acceleration and back again is as simple as differentiating (adding another dot) or integrating (removing a dot). With that notation in hand, Newton was able to create a simple differential equation that describes the motion of all objects in the universe: F = m where F is the force on an object and m is its mass. (Actually, this is not quite a universal law, as the equation only holds when the mass of an object is constant. The more general version of Newton’s law is F = where p is an object’s momentum. Of course, Newton’s equations were eventually refined further by Einstein.) If you’ve got an equation that tells you about the force that is being applied on an object, the differential equation reveals exactly how the object moves. For instance, if you have a ball in free fall, it moves in a parabola, while a frictionless spring wobbles back and forth forever, and a spring with friction slowly comes to rest (Figure 28). As different as these outcomes seem, they are all governed by the same differential equation.
Likewise, if you know the way an object moves—whether it be a toy ball or a giant planet—the differential equation can tell you what kind of force is being applied. (Newton’s triumph was taking the equation that described the force of gravity and figuring out the shapes of the planets’ orbits. People had suspected that