The Calculus Diaries - Jennifer Ouellette [6]
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. Mathematics, he knew, was universal, eternal, and to his mind, far more precious than life.
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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