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reached across his desk for the Marchant mechanical calculator.

Bethe said, “It’s twenty-three hundred.”

Feynman started to punch the keys anyway. “You want to know exactly?” Bethe said. “It’s twenty-three hundred and four. Don’t you know how to take squares of numbers near fifty?” He explained the trick. Fifty squared is 2,500 (no thinking needed). For numbers a few more or less than 50, the approximate square is that many hundreds more or less than 2,500. Because 48 is 2 less than 50, 48 squared is 200 less than 2,500—thus 2,300. To make a final tiny correction to the precise answer, just take that difference again—2—and square it. Thus 2,304.

Feynman had internalized an apparatus for handling far more difficult calculations. But Bethe impressed him with a mastery of mental arithmetic that showed he had built up a huge repertoire of these easy tricks, enough to cover the whole landscape of smal numbers. An intricate web of knowledge

underlay

the

techniques.

Bethe

knew

instinctively, as did Feynman, that the difference between two successive squares is always an odd number, the sum of the numbers being squared. That fact, and the fact that 50 is half of 100, gave rise to the squares-near-fifty trick. A few minutes later they needed the cube root of 2½. The mechanical calculators could not handle cube roots directly, but there was a look-up chart to help. Feynman barely had time to open the drawer and reach for the chart before he heard Bethe say, “That’s 1.35.” Like an alcoholic who plants bottles within arm’s reach of every chair in the house,

Bethe had stored away a device for anywhere he landed in the realm of numbers. He knew tables of logarithms and he could interpolate with unerring accuracy. Feynman’s own mastery of calculating had taken a different path. He knew how to compute series and derive trigonometric functions, and how to visualize the relationships between them. He had mastered mental tricks covering the deeper landscape of algebraic analysis—differentiating and integrating equations of the kind that lurk dragonlike in the last chapters of calculus texts. He was continual y put to the test.

The theoretical division sometimes seemed like the information desk at a slightly exotic library. The phone would ring and a voice would ask, “What is the sum of the series 1 + (½)4 + (⅓)4 + (¼)4 + … ?”

“How accurate do you want it?” Feynman replied.

“One percent wil be fine.”

“Okay,” Feynman said. “One point oh eight.” He had simply added the first four terms in his head—that was enough for two decimal places.

Now the voice asked for an exact answer. “You don’t need the exact answer,” Feynman said.

“Yeah, but I know it can be done.”

So Feynman told him. “Al right. It’s pi to the fourth over ninety.”

He and Bethe both saw their talents as labor-saving devices. It was also a form of jousting. At lunch one day, feeling even more ebul ient than usual, he chal enged the table to a competition. He bet that he could solve any problem within sixty seconds, to within ten percent

accuracy, that could be stated in ten seconds. Ten percent was a broad margin, and choosing a suitable problem was hard. Under pressure, his friends found themselves unable to stump him. The most chal enging problem anyone could produce was: Find the tenth binomial coefficient in the expansion of (1 + x )20. Feynman solved that just before the clock ran out. Then Paul Olum spoke up. He had jousted with Feynman before, and this time he was ready. He demanded the tangent of ten to the hundredth. The competition was over. Feynman would essential y have had to divide one by ? and throw out the first one hundred digits of the result—which would mean knowing the one-hundredth decimal digit of ?. Even Feynman could not produce that on short notice.

He integrated. He solved equations taking the spirit of infinite summation into more difficult realms. Some of these perilous, nontextbook, nonlinear equations could be integrated through just the right combination of mental gimmicks. Others could not be integrated exactly. One could plug in numbers, make estimates, calculate a little, make new estimates, extrapolate a little. One might visualize a polynomial expression to approximate the desired curve. Then one might try to see whether the leftover error could be managed. One day, making his rounds, Feynman found a man struggling with an especial y complicated varietal, a nonlinear three-and-a-half-order equation. There was a business of integrating three times and figuring out a one-half derivative—and in the end

Feynman invented a shortcut, a numerical method for taking three integrals at once and a half integral besides, al more accurately than had been thought possible. Similarly, working with Bethe, he invented a new and general method of solving third-order differential equations. Second order had been manageable for several centuries. Feynman’s invention was precise and practical. It was also doomed to a quick obsolescence in an age of machine computation, as was, for that matter, the skil of mental arithmetic that did so much to establish Feynman’s legend.

Computing by Machine

Not only the atomic era but also the computer era had its start in those years. Scattered about the nation’s military and civilian laboratories, a few researchers focused exclusively on the means of calculating instead of the calculations themselves. At Los Alamos, in particular, the demand for numerical computation grew more intense than anywhere else on earth. The means were mechanical and now partly electronic, though the crucial technological key, the transistor, remained to be invented at the decade’s end. Calculating technology became a hybrid with machine parts and human parts: people carrying cards from place to place served as the memories and logical-branching units of near computers that stretched across rows and columns of desks.

The bomb project could draw on the best technology

available anywhere, but the best technology offered little to the working scientist. The manufacturers of such equipment

—the International Business Machines Corporation already preeminent among them—considered the scientific market to be negligible. It could not imagine the vast clientele that would soon consume as much calculating capacity as could be created: for forecasting weather, designing engines, analyzing proteins, scheduling airplanes, and simulating everything from ecosystems to heart valves. Business was thought to be the sole potential consumer for business machines, and business meant accounting, which meant addition and subtraction. Multiplication seemed a luxury, although it might be necessary to multiply monthly sales by twelve. Division by machine was esoteric. Computation of mortgage payments and bond yields could be managed by humans with standard tables.

The workhorse of scientific calculating was the Marchant calculator, a clattering machine nearly as large as a typewriter, capable of adding, subtracting, multiplying, and with some difficulty dividing numbers of up to ten digits. (At first, to save money, the project ordered slower, eight-digit versions as wel . They were rarely used.) In these machines a carriage spun around, propel ed at first by a hand crank and later by an electric motor. Keys and levers pushed the carriage left or right. Counter and register dials displayed painted digits. There were rows and columns of keys for entering numbers, a plus bar and a minus bar, a multiplier key and a negative multiplier key, shift keys, and a key for stopping the machine when division went out of control, as

it often did. Mechanical arithmetic was no simple affair.

With al its buttons and linkages the Marchant was not quite as powerful as the giant Difference Engine and Analytical Engine, invented in England a century before by Charles Babbage in hopes of generating the printed tables of numbers