On the last day of 2001, I have been sending thank-you notes. These, for reasons of etiquette that my wife understands but I do not, cannot be typed or generated using a word processor; like letters of condolence, they must be hand-written. That my handwriting over the years has, from lack of use, become cramped and crabbed and illegible to the point where I can barely read it myself is irrelevant. Good luck to the recipients, forced to decipher my script that they see only once a year.

However, I could once write clearly. The proof is available to me, in the form of old correspondence and notes. I wonder about the rising generation, for whom writing with pen and paper may seem as archaic a form of communication as inscribing on clay tablets. I am also led to ponder other obsolete skills.

If calligraphy has become a specialist subject taught in art classes, mental arithmetic is even more of a lost cause. It is not, to my knowledge, taught anywhere at all. I watch and disapprove as my daughter uses an electronic calculator to multiply three and a half by two, while at the same time I recall my parents, who casually checked grocery bills in their heads. They did not consider themselves in any way exceptional. The ability to perform simple arithmetic, written and mental, was taken for granted fifty years ago. To make a stir, you had to possess some truly phenomenal calculating ability.

A few hundred years ago, when computers and automatic calculators did not exist, people with those phenomenal abilities were widely recognized as valuable. Jedediah Buxton, George Parker Bidder, Truman Henry Safford, and Johann Zacharias Dase were calculating wizards who routinely multiplied together in their heads seven-digit numbers, raised integers to high powers, and extracted cube roots. They also performed functions useful to society, solving problems in finance, construction, and cartography that called for masses of numerical results.

They were all, not surprisingly, obsessed by numbers. At a dance, Buxton would count the number of steps taken but otherwise had no interest in the scene. In 1754, he attended a performance by the famous actor David Garrick only in order to count the spoken words. Bidder, who in the 1820s was "rescued" from his life as a "calculating boy" and later became a well-known engineer, said that in his youth numbers were "my friends, and I knew all their relations and acquaintances." The illiterate Dase, a German farmer's son who had no mathematical ability other than calculation, made tables of factors for numbers up to almost eight million, purely for pleasure. He could multiply two twenty-digit numbers in his head in about six minutes, with unfailing accuracy. He also calculated the value of pi correctly to 200 places, a task which he completed in about two months.

Safford did all his calculations mentally, but not without physical side effects. At ten years old, he was asked to square the number 365,365,365,365,365,365. He did so, in a minute or two, but according to an observer, "He flew around the room like a top, pulled his pantaloons over the tops of his boots, bit his hands, rolled his eyes in their sockets, sometimes smiling and talking..."

Safford later became an astronomer, and gradually lost the calculating ability. He also never exhibited his powers in public. Others, however, made national or international tours. Zerah Colburn, the "American lightning-calculator boy," made appearances in England when he lived there between 1810 and 1820. To give an idea of what he could do, he was once asked if 4,294,967,297, a famous number known as the sixth Fermat number, had any divisors. He thought for a little while, then announced that it could be divided exactly by 641. Colburn was right, as anyone today can check in a moment on a calculator that costs just a few dollars. But asked how he knew, he could not explain.

Even a hundred years ago, people with a gift for rapid calculation were esteemed more than today, when a computer sits at most people's elbow. Percy Macmahon, after a career in the British Royal Artillery, took up mathematics late in life, and in 1914, at the age of 61, generated the "enormous, impossible streams of numbers" required for a branch of pure mathematics known as the theory of partitions. Macmahon is the rare exception, a calculating talent who did his work late in life. Most lightning calculators are at their best when young, and to my knowledge all of them have been male.

Occasionally contests were held between competing calculators. I think that this type of entertainment has ended, and I myself witnessed only one. Wim Klein was a Dutch lightning calculator. Alexander Aitken was a New Zealander who for many years served as a professor of mathematics at the University of Edinburgh, and who was also incidentally a calculating wizard. At a "face-off," three things became clear to me, and I suspect to everyone else in the audience. First, each man found trivial such things as multiplying together two four-digit numbers. They gave their answers instantly, as though they already knew them, and perhaps they did. Second, each had his specialties. Klein was better at straight multiplication and division. Aitken, not surprisingly, was better at anything involving prime numbers and factors, where his knowledge of mathematical theorems could be brought into play. And third, both men were so much better than the rest of us, they seemed like members of a separate and superior species.

And now for a question. Despite the evidence of television game shows, I do not believe that people today are less intelligent (however we define that word - see earlier c) than they were 50 or 200 years ago. Nor do I think that extreme abilities, so far removed from the average that they are close to incomprehensible, are less common. If we are no longer producing calculating prodigies, because such powers no longer seem to have value, then in what new directions are talents more appropriate to today's needs being developed?

I do not know. But I bet it involves teenagers, computers, and the Internet; and I suspect you would not like the answer even if I could provide it.

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