Why Beauty is Truth by Ian Stewart Page A

maid, in 1776. Their son Johann Friederich Carl (who later always called himself Carl Friedrich) was born in 1777.
    Gebhard was honest but obstinate, ill-mannered, and not very bright. Dorothea was intelligent and self-assertive, traits that worked to Carl’s advantage. By the time the boy was two, his mother knew she had a prodigy on her hands, and she set her heart on ensuring that he received an education that would allow his talents to flourish. Gebhard would have been happier if Carl had become a bricklayer. Thanks to his mother, Carl rose to fulfill the prediction that his friend, the geometer Wolfgang Bolyai, made to Dorothea when her son was 19, saying that Carl would become “the greatest mathematician in Europe.” She was so overjoyed that she burst into tears.
    The boy responded to his mother’s devotion, and for the last two decades of her life she lived with him, her eyesight failing until she became totally blind. The eminent mathematician insisted on looking after her himself, and he nursed her until 1839, when she died.
    Gauss showed his talents early. At the age of three, he was watching his father, at that point a foreman in charge of a gang of laborers, handing out the weekly wages. Noticing a mistake in the arithmetic, the boy pointed it out to the amazed Gebhard. No one had taught the child numbers. He had taught himself.
    A few years later, a schoolmaster named J. G. Büttner set Gauss’s class a task that was intended to occupy them for a good few hours, givingthe teacher a well-earned rest. We don’t know the exact question, but it was something very similar to this: add up all of the numbers from 1 to 100. Most likely, the numbers were not as nice as that, but there was a hidden pattern to them: they formed an arithmetic progression, meaning that the difference between any two consecutive numbers was always the same. There is a simple but not particularly obvious trick for adding the numbers in an arithmetic progression, but the class had not been taught it, so they had to laboriously add the numbers one at a time.
    At least, that’s what Büttner expected. He instructed his pupils that as soon as they had finished the assignment, they should place their slate, with the answer, on his desk. While his fellow students sat scribbling things like
    1 + 2 = 3
    3 + 3 = 6
    6 + 4 = 10
    with the inevitable mistake
    10 + 5 = 14
    and running out of space to write in, Gauss thought for a moment, chalked one number on his slate, walked up to the teacher, and slapped the slate face down on the desk.
    â€œThere it lies,” he said, went back to his desk, and sat down.
    At the end of the lesson, when the teacher collected all the slates, precisely one had the correct answer: Gauss’s.
    Again, we don’t know exactly how Gauss’s mind worked, but we can come up with a plausible reconstruction. In all likelihood, Gauss had already thought about sums of that kind and spotted a useful trick. (If not, he was entirely capable of inventing one on the spot.) An easy way to find the answer is to group the numbers in pairs: 1 and 100, 2 and 99, 3 and 98, and so on, all the way to 50 and 51. Every number from 1 to 100 occurs exactly once in some pair, so the sum of all those numbers is the sum of all the pairs. But each pair adds up to 101. And there are 50 pairs. So the total is 50 × 101 = 5050. This (or some equivalent) is what he chalked on his slate.
    The point of this tale is not that Gauss was unusually good at arithmetic, though he was; in his later astronomical work he routinely carried out enormous calculations to many decimal places, working with the speed of an idiot savant. But lighting calculation was not his sole talent. What he possessed in abundance was a gift for spotting cryptic patterns in mathematical problems, and using them to find solutions.
    Büttner was so astonished that Gauss had seen through his clever ploy that, to his credit, he gave the boy the