Books: January 22, 1997
DESCRIBING THE WORLD IN NUMBERS
The author explores the wonders and history of calculus
A Tour of the Calculus
David Berlinski *68
Pantheon Books, $27.50
Calculus was invented in the 17th century, lifting a dark shroud that had been blocking the human mind from a clear view of the natural universe. It was created independently by two of the supreme intellects of that age or any age: Isaac Newton, who needed calculus to comprehend gravitation, and Baron Gottfried Wilhelm von Leibnitz, the German mathematician and philosopher who crafted the symbols still used today.
A startling new kind of mathematics that allowed the calculation of an irregular area along with the slope of any curve, calculus was the first significant step beyond Greek geometry. (Eventually, calculus also provided for the calculation of changes in variables in relationship to one another as well as the maximum and minimum values of functions.) Recognizing the importance of the new system, which would open the way to modern science by allowing us to translate natural cycles such as the motion of the planets and sun into numbers, Leibnitz and Newton engaged in a bitter, lifelong dispute over who was first.
In writing this beguiling yet often vexing book, David Berlinski *68's avowed purpose is to bring to his readers a sense of calculus as a profound mental tool of almost unbelievable power. Remarkably, the author, a mathematician, former teacher, and systems analyst who now writes full time in San Francisco, accomplishes a lot more. In fact, his book could just as well have been entitled "A Brief Tour of Human Thought: Or How I Learned to Love Calculus."
Berlinski's tour begins with a visit to Zeno the Eleatic, a Greek who lived in the fifth century b.c. In his famous paradox, Zeno argued that, mathematically, it was impossible for a person to cross a room. "How so?" wonders the author with the assurance of one who knows and is about to enlighten the uninitiated. In order to reach the far wall, according to Zeno, a person must first cross half the room, then half the remaining distance, and then half of what still remains, ad infinitum. Such a trip, Zeno argued, would go on indefinitely.
From Zeno's "diamondbright little argument," which was really an explication of the Greek struggle to mathematize nature, we jump forward to the 17th century. Before then, writes Berlinski, "everything is squid ink and ocean ooze and dark clotted intuitions; but afterward a strange symbolic system erupts into existence and floods the intellectual landscape with a hard flat nacreous light." This system is calculus and the key to its discovery was understanding velocity. The author imagines a motorcyclist streaking across the desert. The driver is pulled over for speeding and argues with the patrolman that he was really averaging only 55 miles per hour-not the 80 miles per hour detected by radar. The patrolman, of course, is not interested in average velocity, but in the driver's speed at a certain instant.
But an instant has no time. So how can there be such a thing as velocity at an instant since the very definition of velocity is distance over time? This problem was at the heart of the discovery of calculus. Grappling with it, writes Berlinski in a wonderfully imaginative description, was "the disheveled and half dressed Newton, his crumbfilled wig askew, shambling about the evil-smelling room in which he lived and worked, muttering to himself, his thin lips half forming words . . . forgetting to eat and sleeping in weak, disorganized fits, an apple rotting on the desk."
Chapter by chapter, Berlinski explores the wonders of calculus. On one occasion he demonstrates the difference between real and imaginary numbers by means of an argument with an irate cab driver; he illustrates continuous functions in terms of a leisurely walk around Prague; and in a passage from Anna Karenina he finds the essential difference between a novelist and a mathematician. But eventually the going gets tough. And by the middle of the book he will have lost most of his mathematically disinclined readers with passages that tax their ability to work through the logic of theorems.
Sometimes the author borders on the precious, striving too hard for effect or, he cheerfully admits, for the sheer joy of listening to himself talk (or write). In evoking a visit to Prague where he gave a lecture, he mocks the English accents of his hosts-"You lif in California, tak?"-and in another case that of a brilliant Russian émigré-"Iss not possible forget"-perhaps to demonstrate that he has a fine ear for dialect.
In the work of Leibnitz, the author sees the concept of velocity and, hence, calculus coming into focus. He imagines a portly man sitting in a brownwalled room in Hanover, Germany. Leibnitz allows his thoughts to roam through the great room of his mind. As Berlinski describes it, "The steady rhythmic ticking of the clock prompts Leibnitz to imagine the sounds suspended in space: he seems to see what he hears, each gentle tick breaking open before his eyes like a small multicolored burst. He attends to one such burst, the sound (or the sight) marking the moment as it arises and then perishes, and with the thumb and forefinger of his left hand, he measures the distance between that sound, which has already vanished, and the one to come." Leibnitz wonders what happens if the time between clock ticks becomes too small for his fingers to measure.
"How small? Leibnitz asks himself."
"With great delicacy, Leibnitz allows his forefinger to approach his thumb. Very small. He can sense in the skin between his fingers the contracted space between his flesh. Infinitely small." And so, the author imagines, calculus was born.
With writing this good, most readers probably will not worry that Berlinski's goal of making modern calculus accessible to anybody with the patience to follow his playful and witty arguments remains elusive.
-John Boslough '67
John Boslough's latest book, The Very First Light: The True Inside Story of the Scientific Journey Back to the Dawn of the Universe, with coauthor John C. Mather, was published this month by Basic Books.
THE MYSTICAL DIMENSIONS OF ANGLING
Fly Fishing: A Life in Mid-Stream
Turhan Tirana '57
Kensington Books, $20
Turhan Tirana '57's collection of essays on fly fishing is eclectic and unpretentious, and discursive in the best tradition of fish stories. He recalls with vivid detail his first-ever fish, a yellow perch caught on a Nantucket farm pond when he was nine. He reflects on the charms and pitfalls of fishing with a spouse and children. He writes about the history of a pastime first chronicled 1,700 years ago by the Roman naturalist Aelian, who described Macedonians fishing for trout with a hook wound with scarlet wool and a cock's feathers; and about the sport's current cachet, in part the result of Robert Redford's movie A River Runs Through It. He plumbs the psychology of our desire to catch fish and the mystical and even religious dimensions of fishing. He looks at the Ghost of Angling Future and imagines himself fishing in his dotage, like the 90-year-old blind physician he once knew who cast from a boat with a guide at his side.
As a genre the angling essay is notoriously self-indulgent, and there were times when I wished the author didn't stray so far from his own experience (he writes about fishing in Slovenia and Macedonia, where he's actually wet a line, but also about Norway and New Guinea, where he hasn't). Tirana wrote this book in his late 50s, old enough for him to have withered into a truth or two about fishing. "Angling," he notes, "presupposes a desire to catch fish." But after passing through the well-documented phases of fish-catching (landing the most, or the biggest, or the hardest-to-fool), we may be wise or lucky enough to realize that the pursuit is its own reward, and "success is the release from the need for more success."
-J. I. Merritt '66
BOOKS RECEIVED
Dauntless Marine: Joseph Sailer, Jr. ['30], Dive-Bombing Ace of Guadalcanal
Alexander S. White
White Knight Press, $24.95
Have Another God Day
Philip Humason Steinmetz '30
Orders to Philip Humason Steinmetz, 454 Maple Ave., Marietta, GA 30064-2114. $5 paper
Carlos Lacerda, Brazilian Crusader: Volume II: The Years 1960-1977
John W. F. Dulles '35
University of Texas Press, $50
The Corsair Years
(World War II memoir)
Andrew Jones '44
Turner Publishing, $24.95
The Captive of Truth (novel)
Jack Boice '45
Pentland Press, 5124 Bur Oak Circle, Raleigh NC 27612. $16.95 paper
To the Poles by Ski and Dogsled
Joseph E. Murphy '52
Crossgar Press, 2116 West Lake of the Isles Parkway, Minneapolis, MN 55405. $9.95 paper
The Compromises Will Be
Different (poems)
Robert Rehder '57 *70
Sheep Meadow Press, $12.95 paper
Navajo
Suzanne and Jake Page '58
Harry N. Abrams, $49.50
Old Barns in the New World: Reconstructing History
Richard W. Babcock and
Lauren R. Stevens '60
Berkshire House Publishers,
$16.95 paper
Keeping an Open Door: Passages
in a University Presidency
Keith Brodie '61 and Leslie Banner
Duke University Press, $22.95
Heroes of the Age: Moral Fault Lines on the Afghan Frontier
David B. Edwards '75
University of California Press,
$50 cloth, $20 paper
Psychotherapy and Buddhism: Toward an Integration
Jeffrey B. Rubin '75
Plenum Publishing Corporation, $39.50
Through Loona's Door: A Tammy and Owen Adventure with Carter G. Woodson (young readers)
Tonya Bolden '81,
Illustrated by Luther Knox
Corporation for Cultural Literacy, 130 Webster St. #100, Oakland, CA 94607. $15.95