Nick Tussing
Biography Paper
Jahann Carl Friedrich Gauß (1777-1855)
Jahann Carl Friedrich Gauß was one of the world's greatest mathematicians, scientists, and men
of genius. He was born on the 30th of April 1777. From the age of three his genius was apparent. It is
said that when he was three years old he was sitting with his father and corrected an arithmetic error he
had made while figuring money. In primary school Gauß's teacher though he would occupy the class
by having them add the numbers 1 through 100 together. Gauß intuitively realized that this was a sum
of a series and created a formula to add them all together. He did this by realizing that through
pairwise addition of the first and last, the second and the second from last, and so on, would yield 101.
Thus the solution to the problem would be 101 * 50 = 5050. He produced this answer and formula in
seconds (http://www.geocities.com/RainForest/Vines/2977/gauss/english.html).
Gauß was awarded a fellowship to the Collegium Carolium at the age of 15 by the Duke of
Brunswick. While in college Gauß devoted a lot of his time to rediscovering theorems on his own.
This exercise lead him to his first huge discovery. In 1796 Gauß proved that “any regular polygon with
a number of sides which is a Fermat prime (and, consequently, those polygons with any number of
sides which is the product of distinct Fermat primes and a power of 2) can be constructed by compass
and straightedge” (http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss). The most noted of these
Fermat primes is 17. The construction of the regular heptadecagon had eluded the Greeks, and
everyone who had studied constructions thereafter. This breakthrough had been the only significant
advancement in the field of geometric constructions in centuries (http://www-history.mcs.standrews.ac.uk/Biographies/Gauss.html).
Later that same year Gauß invented modular arithmetic. This intuitive method of analyzing
numbers make many manipulations in mathematics possible, and much easier. The very basis of things
such as the workings of the computer would not work without modular arithmetic. The use of modular
arithmetic lead to his proof of the quadratic reciprocity law. This was the first ever proof of the law.
Gauß, later on in that year, made an important conjecture about the Prime Number Theorem. Using the
offset logarithmic integral function, Gauß put forth the most accurate approximation to the prime
counting function to that point in time. This work was very influential in the work of both Chebyshev
and Reimann's work on the subject (http://en.wikipedia.org/wiki/Prime_number_theorem). Towards
the end of 1796, Gauß discovered that all positive integers are the sum of no more than 3 triangular
numbers. He also published a work on the number of solutions of polynomials in finite fields. This
work would be very influential in Andre Weil's work on generating functions in the late 1900's
(http://en.wikipedia.org/wiki/Weil_conjectures).
In the year 1799 Gauß published a proof of the fundamental theorem of algebra. In fact, he
published 4 different proofs of the theorem. These 4 proofs were done over the span of his life. His
original proof was not accepted within the mathematical community. He tried throughout his life to
improve on this proof, and his last and final proof is quite rigorous, even by today's standards. Gauß
was a perfectionist, and often would not publish if his works were, in his mind, arguable in any way.
This lead him to not publish many works throughout his life that would be years ahead of their time.
He had a matto about his works, “Few but ripe” (http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss).
In 1801 Gauß proved the fundamental theorem of arithmetic in his book Disquisitiones
Arithmeticae.
In this same book he presented a clear presentation of modular arithmetic. This book
is known as a monumental work in the field of number theory, and shaped the field into what it is today
(http://scienceworld.wolfram.com/biography/Gauss.html). It was in that year that Gauß would turn his
attention from math, to the sciences. An asteroid named Ceres came into view for just a few days in
that year, and was observed by an Italian astronomer named Giuseppe Piazzi. Using Piazzi's
observations, Gauß was able to predict where and when Ceres would be visible again. Gauß choose to
search for a position in astronomy. In 1807 he was appointed Professor of Asttronomy in Gottingen.
Gauß remained in the position for the remainder of his life.
This discovery of Ceres would eventually lead to one of Gauß's greatest works. After three
months of tracking the planetoid, it was lost to astronomers behind the glare of a star. Astronomers
were unable to find it again. Gauß set out to find out where to find Ceres once again. It took him three
months of calculations and formulations, but he was able to tell astronomers where to find Ceres again.
His prediction was amazingly accurate, given the precision of tools at the time. A few years later Gauß
published a book about the work that he did, called “Theory of motion of the celestial bodies moving in
conic sections around the sun” (http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Gauss.html).
This work contained one of the most rigorous astronomical calculations ever documented and is
considered a cornerstone of astronomical computation. In this book Gauß defined the Gaussian
gravitational constant.
In determining the constant, Gauß used a treatment of the method of least squares, a treatment
that was previously not widely used, and is now a widely used method of containing error. Gauß
himself proved the method in 1809, assuming normally distributed errors. The method was described a
few years before by Adrien-Marie Legendre, however Gauß claimed to have been using it for nearly 15
years, but did not want to publish until he had proved it. The method of least squares is used in many
branches of math and science (http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss).
Over the next ten years of his life Gauß had little work published and did not come out with any
new breakthroughs. It is said that he researched lightly into the idea of non-Euclidean geometries but
never published anything about it. A friend of his, Bolyai, is credited with discovering non-Euclidean
geometry. A feat that Gauß refused to praise, saying that it was nothing that he had not already
understood for the past 35 years (http://www-groups.dcs.st-and.ac.uk/~history/Biographies/
Gauss.html). In 1818 Gauß accepted a commission from the state of Hanover to carry out a geodetic
survey in order to link up with the already exisisting Danish grid. This commission lead him to
develop yet another amazing breakthrough. Through manipulation of the data, Gauß created the
Guassian distribution, which is today known as the Normail distribution. This distribution is one of the
foundations of modern statistics and one of the most used tools of mathematics within the sciences.
Another 10 years later Gauß published the “remarkable theorem,” which described curvature
within the field of differential geometry. This is an important theorem that says that the curvature of a
surface can be determined entirely by measuring angles and distances on the surface; that is, curvature
does not depend on how the surface might be embedded in (3-dimensional) space
(http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss). The remainder of his life's work dealt with the
scientific field of magnetism. Gauß struck up a friendship with Wilhelm Weber the physics professor
at the University, and they together found a few advancements in magnetism which would influence
the work of Kirchhoff, and help lead to Kirchhoff's law of circuits. Gauß died in 1855, at the age of 78.
His brain was preserved and studied, in hope of finding a “cause” of his genius. It was said to be large,
and contained highly developed convolutions (Dunnington, 1927).
Gauß put his work before his family life. He did not get along well with his father, who
wanted him to be a mason. His mother pushed him to excell, and later in life continued to live with
him once his father had died. Gauß married twice in his life, and had three children with both. Gauß
was haunted by the death of his first wife, and his daughter in 1809. Gauß's second marriage was said
to be an unhappy one. Gauß did not get along well with any of his sons, who all left his house, and two
of which moved to the United States. Over his life time Gauß influenced hundreds of scientists and
mathematicians. He had groundbreaking discoveries in at least 4 fields of mathematics, physics, and
astronomy. There are at least 25 theorems, techniques, constants, or methods named after Gauß and at
least 30 more with the name Gaussian in their title. Gauß is accepted as one of the most influential
people who ever lived, and is to be inducted this year into the Walhalla Temple
(http://en.wikipedia.org/wiki/Walhalla_temple).
Bibiolography
Dunnington, G. Waldo. (May, 1927). "The Sesquicentennial of the Birth of Gauss". Scientific Monthly
XXIV: 402–414.
http://www.geocities.com/RainForest/Vines/2977/gauss/english.html. “ Johann Carl Friedrich Gauss.”
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Gauss.html. “Johann Carl Friedrich Gauss.”
1996. JCR/EFR.
Weisstein, Eric W. http://scienceworld.wolfram.com/biography/Gauss.html. “Carl Friedrich Gauss
(1777-1855).” 1996.
Wikipedia. http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss. “Carl Friedrich Gauss.” 2007.
Wikimedia Foundation INC.
Wikipedia. http://en.wikipedia.org/wiki/Prime_number_theorem. “Prime Number Theorem.” 2007.
Wikimedia Foundation INC.
Wikipedia. http://en.wikipedia.org/wiki/Walhalla_temple. “Wakhalla Templ.” 2007. Wikimedia
Foundation INC.
Wikipedia. http://en.wikipedia.org/wiki/Weil_conjectures. “Weil Conjectures.” 2007. Wikimedia
Foundation INC.