A Biography of Carl Friedrich Gauss
(Born: April 30, 1777, Brunswick, Germany - Died: February 23, 1855, Gottingen, Germany)
Synopsis
Carl Friedrich Gauss, originally named Johann Friedrich Carl Gauss, was a German born mathematician, physicist, and inventor that is regarded as one of the greatest mathematicians of all time for numerous contributions to number theory, geometry, probability theory, geodesy, astronomy, the theory of functions, and physics, particularly the study of electromagnetism.
Carl Friedrich Gauss, 1803
Early Life
Gauss was an only child born into a poor, working-class family. His father, a gardener and brick-layer, had a reputation as a standup, honest man, but he discouraged young Carl from attending school, with expectations that his son would follow in one of the family trades. Luckily, Gauss' mother and uncle saw potential in his genius early on and encouraged him to embrace his intelligence and reinforced it with a good education. He was unique among mathematicians in that he was a calculating prodigy, and retained a knack for performing elaborate calculations in his head through his adult years.
At the age of three, he is said to have corrected an error in his father's payroll calculations, and by the age of 5 he was looking after his father's accounts on a regular basis. Later in life, he joked that he could calculate before he could speak. While in math class, at the age of ten, Gauss was tasked with finding the sum of every number from 1 to 100. The teacher expected the beginner's class to take a reasonable amount of time to complete the exercise, but in a few seconds, Carl proceeded to the front of the classroom and placed his slate on the teacher's desk. On the slate was one number, 5,050. Carl then went on to explain to his teacher that 1+100= 101, 2+99=101, 3+98=101, etc. Also, there were 50 pairs of numbers that each added up to 101. Thus, 50 times 101 equals 5,050.
At fifteen, Gauss recognized a pattern in the occurrence of prime numbers, a problem which had entertained the minds of the best mathematicians since ancient times. Although the occurrence of prime numbers appeared to be completely random to some, Gauss approached the problem by graphing the number number of primes less than a given number. He noticed a rough trend: as the numbers increased by 10, the probability of prime numbers occuring reduced by a factor of approximately 2. For example, there is a 1 in 4 chance of getting a prime from 1 to 100, a 1 in 6 chance of a prime number from 1 to 1,000, a 1 in 8 chance of a prime number from 1 to 10,000, etc. However, he was aware that this method yielded an approximation and as he couldn't provide a proof of his findings, he kept his theory secret until much later in life.
College Years
Carl was also gifted with regards to languages, his teachers and mother recommended him to Carl Wilhelm Ferdinand, Duke of Brunswick in 1791. After meeting Gauss, the Duke was so impressed by the gifted student with a photographic memory that he who him financial assistance to pursue an education and then to study mathematics at the University of Gottingen from 1795 to 1798. At the end of his college years, Gauss made his first mathematical discovery that, until then, was thought to be impossible. He found that a regular polygon with 17 sides could be drawn using a compass and straight edge. Its significance lies not in the result, but in the proof, which relied on an in-depth analysis of the factorization of polynomial equations and paved the way for later ideas related to Galois theory. This proof represented the first advancement in regular polygon construction in over 2,000 years.
A visual representation on constructing a heptadecagon
Pleased with this discovery, Gauss abandoned his study of languages in order to focus solely on mathematics. His doctoral thesis provided a prooof of the fundamental theorem of algebra: every polynomial equation with real or complex coefficients has as many roots or solutions as its degree or the highest power of the variable. Gauss' Proof, though not entirely convincing, was remarkable for its critique of previous attempts. Later, Gauss generated three more proofs of this major result, which shows the importance of the topic.
Career
Disquisitiones Arithmeticae
Gauss' notoreity as a truly remarkable mathematician resulted from two major publications in 1801. The first of which was his publication of the first systematic textbook on algebraic number theory, Disquisitiones Arithmeticae. It opens with the first account of modular arithmetic, provides a thorough account of the solutions of quadratic polynomials in two variables in integers, and finishes with the theory of factorization previously mentioned. The material and its natural generalizations set the stage in number theory for much of the 19th century, and Gauss' continuing interest in the subject inspired much more research, especially in German universities.
The second publication was related to his rediscovery of the asteroid Ceres. Its original discovery, by the Italian astronomer Giuseppe Piazzi in 1800, but it disappeared behind the Sun before oberservers could calculate its orbit with adequate precision to know where it would reappear. Although a myriad of astronomers competed for the honor of finding it again, Gauss succeeded by devising a novel method for dealing with errors in observations. The method is now known as the method of least squares and is one of several contributions to the field of probability and statistics. Carl went on to work for many years as an astronomer and published another major work on the computation of orbits. Eventually, these discoveries led to Gauss' appointment as professor of mathematics and director of the observatory at Gottingen.
Similar mathematical motives led Carl to accept the challenge of surveying the territory of Hanover, and he was often out in the field in charge of collecting data and making observations. The project spanned from 1818 to 1832 and encountered numberous difficulties, but led to a series of advancements. One such development was Gauss' invention of the heliotrope, which is an instrument that reflects the Sun's rays in a focused beam that can be observed from several miles away, thus improving the accuracy of observations. Another discovery showed that there's an intrinsic measure of curvature that is not altered if the surface is bent without being stretched. For example, a circular cylinder and a flat sheet of paper have the same intrinsic curvature, which is why exact copies of images on a cylinder can be represented on paper. However, a sphere and a plane have different curvatures, which is why no complately accurate flat map of the Earth can be produced.
Gauss' Heliotrope
In the 1830's, Gauss became interested in terrestrial magnetism and participated in the first worldwide survey of the Earth's magnetic field. To assist the project, he invented the magnetometer, which was the first capable of measuring the absolute magnetic intensity. Working with his Gottingen colleague, the physicist Wilhelm Weber, they made the first electric telegraph. Once again, important mathematical discoveries were made in the field of potential theory, an branch of mathematical physics arising in the study of electromagnetism and gravitation. The two are also credited with the independent discovery of Kirchhoff's circuit laws in electricity.
He also made contributions to cartography, the theory of map projections. For his work related to the study of angle-preserving maps, he was awarded a prize from the Danish Academy of Sciences in 1823. This work suggested that complex functions of a complex variable are generally angle-preserving, but wasn't explicitly stated until it was investigated by Bernhard Riemann, who was inspired by Gauss. He also had additional unpublished theories into the nature of complex functions and their integrals, some of which he discussed with friends.
Gauss had perfectionist tendencies and would often withhold publication of his theories and discoveries. As a student, he began to doubt the a priori truth of classical Euclidean geometry. For this to be the case, there must exist an alternative geometric description of space. Rather than publish such a theory, Gauss resorted to criticizing various a priori defenses of Euclidean geometry and theorizing that there exists a logical alternative to Euclidean geometry. Two mathematicians, the Hungarian Janos Bolyai and the Russian Nikolay Lobachevsky published their description of a novel, non-Euclidean geometry. Some have attributed the failure of Gauss to express his ideas due to his innate conservatism, others to his inquisitiveness that always drew him on to the next new idea, still others to his failure to find a central description that would govern geometry once Euclidean geometry was no longer unique.
Gauss also concealed his ideas from colleagues related to elliptic functions. He published an account of a peculiar infinite series, and he wrote but didn't publish a description of the differential equation that the infinite series satisfies. He demonstrated that the hypergeometric series, can be used to a multitude of familiar and new functions. But by then he knew how to utilize the differential equation to produce a very general theory of elliptic functions and to free the theory from its origins in the theory of elliptic integrals. This was a major discovery, because the theory of elliptic functions naturally treats them as complex-valued functions of a complex variable, but the contemporary theory of complex integrals was insufficient for the task. When some of his theory was published by the Norwegian Niels Abel and the German Carl Jacobi in 1830, Gauss remarked that Abel had come one-third of the way. Although this comment was accurate, Gauss continued to withhold publication and left his work to be rediscovered by others.
Death and Legacy
Gauss passed away peacefully in his sleep in Gottingen on February 23, 1855. He lived to the ripe old age of 77. He was buried in Gottingen's Albanifriedhof Cemetery, close to the university that he spent many years, not only as a student, but as a professor, too. His brain was preserved and still resides in Gottingen's physiology department. In his later years, Gauss remained so proud of his youthful heptadecagon achievement that he requested for the shape to be carved on his tombstone, just as Archimedes had a sphere inside a cylinder carved on his. Unfortunately, his wish was not fulfilled with the stonemason citing that it would be too difficult to carve a heptadecagon that didn't resemble a circle. However, his home town of Brunswick erected a memorial in Gauss' honor complete with a heptadecagon inscribed on it.
