I explained this to Allyson, who responded with an incredulous, “That’s it? Why can’t math teachers just say that?” In fairness, they probably do; they just say it in a foreign language. Galileo famously observed, “Nature’s great book is written in mathematical symbols.” Unfortunately, to the untrained eye and ear, that language resembles ancient Sanskrit, and math teachers may as well be speaking gibberish. Most of us never get past the strange symbols and jargon, and thus meander through life without any quantitative tools beyond basic arithmetic. We can balance a checkbook, but have no grasp of statistics, compound interest, or probability, for example—and this puts us at the mercy of those who do understand them, and thus can manipulate us at will. Knowledge is power, and we forfeit that power when we choose to remain willfully ignorant.
The inability to grasp basic algebra and calculus also can be a stumbling block to many students who otherwise would wish to become scientists. Take my friend Lee, whose struggles with algebra in high school—despite top grades in all her other classes—reduced her to tears, and kept her from becoming a marine biologist. She still loves science, but has a visceral hatred of mathematics to this day. “It wrecked my self-confidence in a way nothing else ever did, and still knots my stomach,” she told me. “I’m not totally innumerate, but anything that looks like an equation makes me break out into a cold sweat and run screaming in the other direction.”
Ironically, given my distaste for the subject, I succeeded at math, at least by the usual evaluation criteria: grades. Yet while I might have earned top marks in geometry and algebra, I was merely following memorized rules, plugging in numbers and dutifully crunching out answers by rote, with no real grasp of the significance of what I was doing or its usefulness in solving real-world problems. Worse, I knew the depth of my own ignorance, and I lived in fear that my lack of comprehension would be discovered and I would be exposed as an academic fraud—psychologists call this “impostor syndrome.”
I might have gone through the rest of my life cringing compulsively at the mere sight of an equation. But I became a science writer and fell in love with physics—not the math part, mind you. I loved the rich history, the people, the funky experiments, and the big ideas. One fateful day, I asked a physicist named Alan why it was true that all objects fall at the same rate, regardless of mass, when casual observation would seem to indicate the opposite. It seemed counterintuitive to me.
This is the basis of a famous experiment proposed by Galileo. If you drop a coin and a feather under normal (atmospheric) conditions, the coin will hit the ground first. But Galileo reasoned that another force—air resistance—slowed down the feather’s descent because it had more surface area than the coin. In a vacuum, there would be no air resistance, so all objects would accelerate equally. He didn’t have the techniques for creating a vacuum back then to test his hypothesis, but Isaac Newton derived Galileo’s assertion mathematically in the seventeenth century.
Today, vacuum technology is commonplace, and the coin-and-feather experiment is a staple of physics demonstrations. I had witnessed one such demonstration, so I knew from experience it was true that objects fall at the same rate regardless of mass—or did I? I hadn’t built and performed the experiment myself. How could I know it wasn’t some kind of trick, or a mistake in the experimental setup?
Alan pondered a moment, stroking his beard, and then pointed out that I need not take the matter on faith. It would become obvious to me why this was so if I allowed him to walk me through the equation.
I resisted. Alan persisted: “It’s not real math; it’s just algebra.” He wore me down eventually, and he was right. On his office whiteboard, he patiently demonstrated how the little m—for the mass of the object, as distinct from a big M for the mass of the Earth—on each side of the equation effectively cancels out, making an object’s mass irrelevant to the rate of acceleration. And I had my first epiphany that math might actually be relevant to my life: Among other advantages, mathematics can help us better grasp the more counterintuitive notions in physics. It is certainly possible to do so without the benefit of algebra or calculus, but when I saw that equation worked out, some final piece of insight clicked into place that I hadn’t even realized was missing. Math finally had a meaningful context.
So began a gradual lowering of my knee-jerk defenses against numbers and abstract symbols, and the start of a grudging appreciation for the role of mathematics in the “real world.” It was a tantalizing glimpse into a whole new way of looking at reality, and for the first time in my mathephobic life, I wanted to learn more. Armed with a few books,3 a DVD lecture series from the Teaching Company, and the support of my physicist spouse, I set out to discover what I’d been missing all those years.
Once I started delving into calculus, I realized that this seemingly arcane subject is applicable to everything from gas mileage, diet and exercise, economics, and architecture to population growth and decline, the physics behind the rides at Disneyland, the probabilities associated with shooting craps in a Vegas casino—even the I Ching. In fact, one could argue that we all do some form of calculus all the time, without realizing it. A baseball outfielder has to estimate where the ball is likely to land after the batter hits it. Whether he knows it or not, his brain is calculating the trajectory of that ball, then sending a signal telling the outfielder where to place himself in order to make the catch. Lurking somewhere in that process is a calculus problem. Or two.
Even lowly worms do calculus, according to a University of Oregon biologist named Shawn Lockery. He’s studied roundworms to figure out how they use their sense of taste and smell to navigate as they forage for food. He compares the approach to the game of hot-and-cold one might play with a child, in which one says, “You’re getting warmer (or colder)” to help said child home in on the target. Roundworms do this, too, changing direction in response to feedback, but they get their feedback by calculating how much the strength of different tastes—in this case, salt concentrations—is changing. In calculus terminology, the worms take a derivative to figure out how much a given quantity is changing at a certain point in space and time, and adjust their behavior accordingly.
If worms can do calculus, human beings simply have no excuse for avoiding it. I think scientists have a valid point when they bemoan the fact that it’s socially acceptable in our culture to be utterly ignorant of math, whereas it is a shameful thing to be illiterate. We could all be just a little bit mathier. We don’t all need to become mathematical prodigies, but we ought to have some basic understanding of how math in general, and calculus in particular, fits into our cultural framework, and be able to look at a rudimentary equation without breaking into a cold sweat. It is an integral part of our intellectual history, after all.
William Benjamin Smith, a math professor in the late nineteenth century, observed in the preface to his book Infinitesimal Analysis, “Calculus is the most powerful weapon of thought yet devised by the wit of man.” Far from being some static, dead set of rules to be memorized and blindly followed, calculus is almost an organic entity. Watch any physicists work a problem, and you’ll see its extraordinary flexibility: They play fast and loose with the numbers, simplifying and rounding up as needed to complete the task at hand. They adapt calculus to their needs—not the other way around. The act of devising a calculus problem from your observations of the world around you—and then solving it—is as much a creative endeavor as writing a novel or composing a symphony. Those things are not easy, nor should they be. As with any art form, the best way to learn and improve is by diligently practicing that art.
In college, I proudly flaunted my mathematical ignorance by sporting a T-shirt reading, “English major—you do the math.” I never realized, until much later, that this defensive, belligerent attitude stood in the way of acquiring genuine understanding. I have a new T-shirt now to symbolize my change in mind-set: “You mess with calculus, you mess with me.” Archimedes certainly felt that way about his geometric diagram. Mathema
tics, he knew, was universal, eternal, and to his mind, far more precious than life.
1
To Infinity and Beyond
You take a function of x and you call it y, Take any x-nought that you care to try, Make a little change and call it delta-x, The corresponding change in y is what you find nex’, And then you take the quotient and now carefully Send delta-x to zero, and I think you’ll see That what the limit gives us, if our work all checks, Is what we call dy / dx, it’s just dy / dx.
—TOM LEHRER, “The Derivative Song”
You never know what you’ll find moldering in a musty old attic: forgotten photo albums, moth-eaten vintage clothing, discarded toys—or maybe a rare mathematical manuscript disguised as a humble prayer book. That’s what one French family discovered in the closet one day in the late 1990s: a battered and smudged prayer book with the faint outlines of Greek lettering in the margins, along with an occasional diagram. Sensing a potentially significant find, the family brought the book to Christie’s auction house of London for appraisal. It proved a financially astute move: in 1998 the prayer book sold for $2 million.
What was so special about a tattered ancient prayer book? It took numerous scientists, digital photography under different wavelengths of light, and a spot of x-ray fluorescence imaging to fully decipher the mystery.4 Almost a decade of intensive scientific analysis revealed that lying just under the surface text of those prayers are the scribblings of Archimedes. Not just random musings, either: It was two lost texts by the great mathematician, including one, entitled The Method, that constitutes the earliest known written work on what would later develop into integral calculus.
Archimedes wrote The Method on a scroll of papyrus over two thousand years ago. Eventually someone copied the text onto animal skin parchment and then set the copy aside to molder in a library in Constantinople until 1229 A.D. It was standard practice in medieval times to reuse parchment, so one day, when a monk named Johannes Myronas needed fresh writing materials, he recycled this papyrus, scraping the surface to remove the old ink and copying his prayers over the remains of the original text. No one knows what happened to the prayer book after that, but it ended up in the possession of a Danish philologist named John Ludwig Heiberg in 1908 while he was visiting that Constantinople library. Heiberg was the first to study the not-quite-erased text under a microscope in hopes of deciphering it. He finished an initial (incomplete) transcription, at which point the book disappeared again, until it was uncovered in that dank French closet ninety years later.
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. We refine our calculation by filling in that same area under the curve with a succession of smaller rectangles, determining their areas, and adding those values together. The smaller the rectangles, the more will be needed to fill in that space, and the closer we will get to the actual area. So we do a third iteration, then a fourth, our rectangles getting smaller and smaller each time, until we collapse in fatigue. Eudoxus called this the “method of exhaustion,” and never was a term more apt. Very little is known about Eudoxus, but he studied under Plato in his youth, walking seven miles each way from his home in Piraeus to attend Plato’s lectures, so he was accustomed to exhaustion.
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 gla
ss 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.”
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 2