best arithmetic textbook that money could buy. Within a week, Gauss had gone beyond anything his teacher could handle.
It so happened that Büttner had a 17-year-old assistant, Johann Bartels, whose official duties were to cut quills for writing and to help the boys use them. Unofficially, Bartels had a fascination for mathematics. He was drawn to the brilliant ten-year old, and the two became lifelong friends. They worked on mathematics together, each encouraging the other.
Bartels was on familiar terms with some of the leading lights of Brunswick, and they soon learned that there was an unsung genius in their midst, whose family lived on the brink of poverty. One of these gentlemen, councilor and professor E. A. W. Zimmerman, introduced Gauss to the duke of Brunswick, Carl Wilhelm Ferdinand, in 1791. The duke, charmed and impressed, took it upon himself to pay for Gaussâs education, as he occasionally did for the talented sons of the poor.
Mathematics was not the boyâs sole talent. By the age of 15 he had become proficient in classical languages, so the duke financed studies in classics at the local gymnasium. (In the old German educational system, a gymnasium was a type of school that prepared its pupils for university entrance. It translates roughly as âhigh school,â but only paying students were admitted.) Many of Gaussâs best works were later written in Latin. In 1792, he entered the Collegium Carolinium in Brunswick, again at the dukeâs expense.
By the age of 17 he had already discovered an astonishing theorem known as the âlaw of quadratic reciprocityâ in the theory of numbers. It is a basic but rather esoteric regularity in divisibility properties of perfect squares. The pattern had already been noticed by Leonhard Euler, but Gauss was unaware of this and made the discovery entirely on his own. Very few people would even have thought of asking the question. And the boy was thinking very deeply about the theory of equations. In fact, thatwas what led him to his construction of the regular 17-gon and thus set him on the road to mathematical immortality.
Between 1795 and 1798, Gauss studied for a degree at the University of Göttingen, once more paid for by Ferdinand. He made few friends, but the friendships he did strike up were deep and long-lasting. It was at Göttingen that Gauss met Bolyai, an accomplished geometer in the Euclidean tradition.
Mathematical ideas came so thick and fast to Gauss that sometimes they seemed to overwhelm him. He would suddenly cease whatever he was doing and stare blankly into the middle distance as a new thought struck him. At one point he worked out some of the theorems that would hold âif Euclidean geometry were not the true one.â At the forefront of his thoughts was a major work that he was composing, the Disquisitiones Arithmeticae , and by 1798 it was pretty much finished. But Gauss wanted to make certain that he had given due credit to his predecessors, so he visited the University of Helmstedt, which had a high-quality mathematics library overseen by Johann Pfaff, the best-known mathematician in Germany.
In 1801, after a frustrating delay at the printerâs, the Disquisitiones Arithmeticae was published, with an effusive and no doubt heartfelt dedication to Duke Ferdinand. The dukeâs generosity did not end when Carl left the university. Ferdinand paid for his doctoral thesis, which he presented at the University of Helmstedt, to be printed as the regulations required. And when Carl started to worry about how to support himself when he left university, the duke gave him an allowance so that he could continue his researches without having to be bothered about money.
A noteworthy feature of the Disquisitiones Arithmeticae is its uncompromising style. The proofs are careful and logical, but the writing makes no concessions to the reader and gives no clue about the intuition behind the theorems. Later, he
It so happened that Büttner had a 17-year-old assistant, Johann Bartels, whose official duties were to cut quills for writing and to help the boys use them. Unofficially, Bartels had a fascination for mathematics. He was drawn to the brilliant ten-year old, and the two became lifelong friends. They worked on mathematics together, each encouraging the other.
Bartels was on familiar terms with some of the leading lights of Brunswick, and they soon learned that there was an unsung genius in their midst, whose family lived on the brink of poverty. One of these gentlemen, councilor and professor E. A. W. Zimmerman, introduced Gauss to the duke of Brunswick, Carl Wilhelm Ferdinand, in 1791. The duke, charmed and impressed, took it upon himself to pay for Gaussâs education, as he occasionally did for the talented sons of the poor.
Mathematics was not the boyâs sole talent. By the age of 15 he had become proficient in classical languages, so the duke financed studies in classics at the local gymnasium. (In the old German educational system, a gymnasium was a type of school that prepared its pupils for university entrance. It translates roughly as âhigh school,â but only paying students were admitted.) Many of Gaussâs best works were later written in Latin. In 1792, he entered the Collegium Carolinium in Brunswick, again at the dukeâs expense.
By the age of 17 he had already discovered an astonishing theorem known as the âlaw of quadratic reciprocityâ in the theory of numbers. It is a basic but rather esoteric regularity in divisibility properties of perfect squares. The pattern had already been noticed by Leonhard Euler, but Gauss was unaware of this and made the discovery entirely on his own. Very few people would even have thought of asking the question. And the boy was thinking very deeply about the theory of equations. In fact, thatwas what led him to his construction of the regular 17-gon and thus set him on the road to mathematical immortality.
Between 1795 and 1798, Gauss studied for a degree at the University of Göttingen, once more paid for by Ferdinand. He made few friends, but the friendships he did strike up were deep and long-lasting. It was at Göttingen that Gauss met Bolyai, an accomplished geometer in the Euclidean tradition.
Mathematical ideas came so thick and fast to Gauss that sometimes they seemed to overwhelm him. He would suddenly cease whatever he was doing and stare blankly into the middle distance as a new thought struck him. At one point he worked out some of the theorems that would hold âif Euclidean geometry were not the true one.â At the forefront of his thoughts was a major work that he was composing, the Disquisitiones Arithmeticae , and by 1798 it was pretty much finished. But Gauss wanted to make certain that he had given due credit to his predecessors, so he visited the University of Helmstedt, which had a high-quality mathematics library overseen by Johann Pfaff, the best-known mathematician in Germany.
In 1801, after a frustrating delay at the printerâs, the Disquisitiones Arithmeticae was published, with an effusive and no doubt heartfelt dedication to Duke Ferdinand. The dukeâs generosity did not end when Carl left the university. Ferdinand paid for his doctoral thesis, which he presented at the University of Helmstedt, to be printed as the regulations required. And when Carl started to worry about how to support himself when he left university, the duke gave him an allowance so that he could continue his researches without having to be bothered about money.
A noteworthy feature of the Disquisitiones Arithmeticae is its uncompromising style. The proofs are careful and logical, but the writing makes no concessions to the reader and gives no clue about the intuition behind the theorems. Later, he
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