Carl Friedrich Gauss

Not till long after his death was it known how much of nineteenth-century mathematics Gauss had foreseen and anticipated before the year 1800. Had he divulged what he knew it is quite possible that mathematics would now be half a century or more ahead of where it is. Abel and Jacobi could have begun where Gauss left off, instead of expending much of their finest effort rediscovering things Gauss knew before they were born, and the creators of non-Euclidean geometry could have turned their genius to other things. (E.T. Bell)

It is said that Archimedes knew all the mathematics that was developed up to his time and some more. The same is said of Newton and Carl Gauss. Of the twentieth century mathematicians, this honor goes to Poincare’. Gauss was a veritable prodigy. So much has been written about his genius in mathematics. One of the anecdotes referring to his extraordinary mental computational skills goes back to when he was only three years old. Gauss’s father, Gerhard Gauss, was a brick layer. One Saturday when he was preparing the weekly payroll for his laborers, the young Carl was looking over his shoulders. When his father wrote down the total amount of the payroll at the bottom, Carl chirped up, “Father, the reckoning is wrong; it should be…” True, the total that his father had written down was wrong and the young Gauss was correct. “In later life he (Carl Gauss) loved to joke that he knew how to reckon before he could talk,” (E.T. Bell, Men of Mathematics, p. 221).

Carl Gauss was born to very poor parents on April 30, 1777 “in a miserable cottage at Brunswick (Braunschweig), Germany.” He was given the name of Johannes Friedrich Carl Gauss but he later dropped Johannes and signed his name simply Friedrich Carl Gauss. His father “worked as a gardener, canal tender and a brick layer” and was an extremely harsh man. He wanted his son to follow in his footsteps and was against schooling of any kind. Gauss’s mother, Dorothea, on the other hand, loved his son very much and supported him against the harsh demands of his father. According to Bell (pp. 219-220), “When the wonder child of two, whose astounding intelligence impressed all who watched his phenomenal development as something not of this earth, maintained and even surpassed the promise of his infancy as he grew to boyhood, Dorothea Gauss took her boy’s part and defeated her obstinate husband in his campaign to keep his son ignorant as himself.”

Gauss started school at the age of seven under a teacher, J.G. Buttner, who was almost a brute. He was very harsh with his young pupils and wouldn’t refrain from physical thrashing whenever there was any occasion for disciplining. One day, he assigned a problem of adding all the numbers from 1 to 100 to the kids just to keep them busy for an hour or so. Gauss who had figured out summing the sequences on his own readily produced the answer of 5050 almost immediately. He wrote just the answer on his slate and put it down as was the custom and waited for the others to finish their work. Buttner,
who saw Gauss putting his slate down, thought that Gauss couldn’t possibly work out the problem so quickly, so his answer would surely be wrong. Buttner was very much impressed when he found that Gauss’s answer was correct. Afterwards, his attitude towards Gauss softened and he helped him later on. “ Out of his own pocket he paid for the best textbook on arithmetic obtainable and presented it to Gauss.”

Bell has narrated a similar anecdote but with different numbers. According to him, Buttner asked his pupils to add 81297, 81495, 81693, …, 100899. In this series, the constant incremental difference from one number to the next is 198 and the total number of terms is 100.

Buttner didn’t have adequate knowledge of arithmetic to guide his gifted pupil but his assistant, Johannes Martin Bartels, was inquisitive in advanced mathematics and together they (Gauss and Bartels) worked on some interesting problems. Working on the Binomial Theorem, expansion of (1+x)^n, in which the exponent n is not necessarily a positive integer; he was led to the examination of convergence of infinite series. According to Bell (p. 223), “Gauss’s early encounter with the binomial theorem inspired him to some of his greatest work and he became the first of the ‘rigorists’…The rigor which Gauss imposed on analysis gradually overshadowed the whole of mathematics, both in his own habits and in those of his contemporaries – Abel, Cauchy – and his successors – Weierstrasse, Dedekind, and mathematics after Gauss became a totally different thing from the mathematics of Newton. Euler, and Lagrange.”

After matriculating from school at the age of 14, Gauss entered Caroline College in 1792. While there, he was able to prove the fundamental theorem of algebra. He developed four independent proofs of this theorem in his lifetime. He left the college in 1795 to continue his education at the University of Gottingen. He had by then developed the “method of least squares.” According to Bell, Gauss shared this honor with Legendre who published the method independently in 1806.”His work on the method of least squares led him to the theory of errors. The well-known bell shaped curve is the outcome of his work on normal distribution of errors. His portrait and the bell shaped curve appeared on the German ten-mark banknote from 1989 to the end of 2001 in his honor.

Gauss’s expenses for his education were paid out of a stipend granted him by Duke Ferdinand of Brunswick so his life was free of financial worries and he could devote his full attention to his work. He stayed at Gottingen as a student for three years, According to Bell, “The three years (October, 1795 – September, 1798) at the University of Gottingen were the most prolific in Gauss’s life…The ideas which had overwhelmed Gauss since his seventeenth year were now caught – partly – and reduced to order. Since 1795 he had been meditating a great work on the theory of numbers. This now took definite shape, and by 1798 the Disquisitone Arithmeticaes (Arithmetical Researches) was practically completed.” The Disquistones Arithmetica “contained a clear presentation of modular arithmetic and the first proof of the law of quadratic reciprocity,” (http://academickids.com/encyclopedia/c/carl_friedrich_gauss .html). In 1796, “he had made one of the most important discoveries – the construction of a regular 17-gon (heptadecagon) by ruler and compasses. This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss’s famous work, Disquistones Arithmetica,” (http://www-groups.des.st-and.ac.uk/~history/Printonly/Gauss .html).

At Gottingen, one of Gauss’s teachers was Abraham Kaestner whom he used to ridicule quite often. He called him “a poet among mathematicians and a mathematician among poets.”

He left Gottingen in 1798 without a degree. He submitted a doctoral thesis to the University of Helmstedt, in absentia. His advisor at Helmstedt was Professor Johann Pfaff whom he already knew. In fact, by then Gauss was well known all over Europe for his outstanding work in mathematics. His thesis dealt with the fundamental theorem of algebra.

With the advent of the nineteenth century, Gauss broadened his interests and began doing work in astronomy. In June 1801, Zach, whom Gauss knew, published his work on the orbital positioning of Ceres, a new small planet which was discovered by G. Piazzi on January 1, 1801. Its position was only approximately determined by Piazzi before the planet disappeared behind the Sun. From whatever information was available, Gauss worked out the position of Ceres which differed from the others. When Ceres was rediscovered by Zach in December 1801, it was almost in the same position that was predicted by Gauss.

Gauss returned to Gottingen as “Director of the Gottingen observatory with the privilege – and duty, when necessary – of lecturing on mathematics to university students,”
in 1807. Much of his time was devoted to astronomical work although he found time to do other things also. His contributions to theoretical astronomy ended in 1817 although he continued making astronomical observations. He was asked to conduct a geodesic survey of the state of Hanover to link up with the existing Danish grid. Out of this work arose his interest in differential geometry and he published several seminal papers on it. According to J.J. O’Connor and E.F. Robertson, “Disquisitiones generales circa curvo (1828) was his most renowned work in this field,” (http://www_groups.des.st_and.ac.uk/~history/Printonly/Gauss .html).

He worked in Physics in collaboration with Professor Weber and made significant contributions in several areas including magnetism and optics. He left his imprint on every thing that he worked on. Bell thought it was unfortunate for mathematics that Gauss diverted his attention from it to other things. Had he continued exclusively in mathematics, he would have made many more revolutionary contributions in it. But he did make many contributions to mathematics of which the contemporary world was unaware. He did not publish these contributions but left them hidden in his diary. He wouldn’t publish anything which was not proven beyond any doubt and to his absolute satisfaction. He was absolutely meticulous about the beauty and completeness of a scientific theory. He said, “A cathedral is not a cathedral till the last scaffolding is down and out of sight.” He hated criticism. But criticism wouldn’t leave him.

One of such issues was the discovery of non-Euclidean geometry. In 1831 Farkas Bolayi (who was one of Gauss’s friends) sent to Gauss his son Janos Bolayi’s work on the subject (non-Euclidean geometry). Gauss replied, “to praise it would amount to praising myself. For the entire content of the work.. coincides almost exactly to my own meditations which have occupied my mind for the past thirty or thirty-five years,” according to Wikiquote. According to O’Connor and Robertson, “Again, a decade later, when he was informed of Lobachevsky’s work on the subject, he praised his ‘genuinely geometric’ character, while in a letter to Schumacher, states that he ‘had the same convictions for 54 years, indicating that he had known of the existence of a non-Euclidean geometry since he was 15 years of age..”

Although some doubted Gauss’s assertions and they created bitterness in Lobachevsky’s mind, his diary confirmed Gauss. He indeed was a born genius and only one of his kinds.

Mathematics is called the queen of sciences and Gauss was called the Prince of Mathematics.

(Henrich Wilhelm Matthaus) Olbers drew Gauss’s attention to Fermat’s Last theorem hoping to entice him to work on it and provide a proof. He replied, “I confess that Fermat’s Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of,” (Wikiquote). It was finally proved by Andrew Wiles in October 1994. Wiles proof covered about 150 pages. According to Wiles, “It’s a 20th century proof. It couldn’t have been done in the 19th century…”