Chapter 22: Gauss's Law
Applications

Gauss

There can be no doubt that Carl Fiedrich Gauss was among the most important physicists and the most important mathematician of his time. Indeed, Gauss was arguably the greatest mathematician of any time; only Isaac Newton is frequently mentioned as his peer.

Gauss was born on April 30^th, 1777 in the Duchy of Brunswick, now a part of Germany. He was a child prodigy, and many stories are told of his early mathematical prowess. It is well-documented that he corrected an error in his father's payroll calculations at the age of three, and as an adult he explained that by his recollection he could count before he could talk. Probably the most famous story about young Gauss occured in 1786, when he was nine years old. His teacher, J. G. Büttner, assigned his class the task of adding all of the numbers from 1 to 100. Gauss turned in his slate after only a few seconds, with only the final answer written down. Büttner studiously ignored him until the class had finished, and was astonished to find that Gauss's answer was correct. He asked Gauss how he had arrived at his answer, and Gauss explained: "100+1=101; 99+2=101; 98+3=101, etc., and so we have as many pairs as there are in 100. Thus the answer is 50 X 101 = 5050." Büttner realized at this point that he was not dealing with a typical student, and took steps to assure Gauss's education. He ordered a more advanced book, and put Gauss under the tutelage of his assistant, Johann Christian Martin Bartels (1769-1836), who was also interested in mathematics. It is interesting to note that Bartels went on to become a professor of Mathmatics at Reichenau in Switerland. His career as a mathematician was undistinguished; however, he was apparently an excellent teacher, and his pupils included Lobachevsky in addition to Gauss.

In addition, either Bartels or Büttner introduced Gauss to several influential people, and arranged financial support for him. At first, this amounted to providing a small stipend to Gauss's father for allowing Gauss to spend his evenings studying rather than doing chores at home. Later, this support included such items as logarithmic tables and mathematical instrument case, and ultimately support for his higher education.

By the age of eleven, Gauss had independently derived the binomial theorem in its most general form, and had begun to study (with Bartels) the theory of infinite series. At about that time, a new math teacher took over at Gauss's school. He handed back Gauss's first paper saying that it was "superfluous" that Gauss's should be in his class. Early in 1792, at the age of 14, Gauss entered the Collegium Carolinium, with the support of the Duke of Brunswick. There, he studied both mathematics and the classical languages. He was particularly interested in the works of Newton, Euler, and Lagrange. It was at this time that he discovered the method of least squares for analyzing experimental data. He did not publish that work until 1809, and others (including Legendre) did so before him. At this time, Gauss also described what is still known as the "Gaussian" probability distribution, and discovered the prime number theorem, which predicts the number of prime number less than any given positive integer, x (in the limit of large x). Gauss's conjecture, that this number is given by x/log(x). Was not proven for over one hundred years, after the combined efforts of many important mathematicians. In 1796, at the age of eighteen, Gauss moved on to the University of Göttingen. He received his Ph.D. only three years later, in absentia, from the University of Helmstedt.

From about 1796 onwards, Gauss produced so great a volume of discoveries in mathematics, physics, and astronomy that it is impossible even to list them in an essay of this size. He was among the first mathematicians to use complex numbers, and the description of complex numbers by Hthe geometry of the complex plane. His Doctoral dissertation included the first proof of the fundamental theorem of algebra, which states that any for polynomial equation with complex coefficients there must exist real or complex roots. Perhaps even more fundamental than the result itself, this theorem brought the entire notion of existence proofs to the forefront of mathematics.

Rather than focus on the particulars of Gauss's discoveries, I would like to mention one overiding theme that comes up again and again in writings about Gauss's career. That is Gauss's extraordinary devotion to absolute rigor in his proofs. In particular, his Disquisitiones arithmeticae, published in 1801 is credited as having established the modern standard for precision in mathematical proof. He considered the work of most previous mathematicians (except Newton) to be loose and filled with "half proofs," which he considered essentially worthless. Unfortunately, his proofs, while utterly rigourous, were also extremely difficult to read. He believed that proofs should be as concise as possible, and should be striped of the ideas that motivated the original investigation. He said "I am never satisfied until I have said as much as possible in as a few words" Unfortunately, the difficulty of reading his proofs often kept others from building on his ideas.

A brief Timeline of Gauss's life:

1777 Born, April 30^th, in the Duchy of Brunswick (now Germany).
1792 Enters Collegium Carolinum.
1799 His doctoral thesis includes the first proof of the fundamental theorem of algebra.
1801 Publishes his Disquisitiones Arithmeticae (disquisition on Arithmetic), which is considered to be the founding document of number theory.
1805 Marries his first wife, Johanna Osthoff.
1806 Birth of his first son, Joseph.
1807 Becomes professor of Astronomy and Director of of the Observatory at Göttingen.
1808 Birth of his first daughter, Minna.
1809 Publication of Theoria Motus Corporum Coelestium (theory of the motion of celestial bodies), his principal work in Astronomy.
1809 Birth of his second son, Louis.
1809 Death of his wife.
1810 Death of his son Louis.
1810 Marries his second wife, Minna Waldeck.
1811 Birth of his third son, Eugene.
1812 Published memoir on hypergeometric series.
1813 Birth of his fourth son, Wilhelm.
1814 Published memoir on a method of approximate integration.
1816 Birth of his second daughter, Therese.
1821 Invented the heliotrope, a surveying tool.
1822-26 Conducted a geodetic survey of the Kingdom of Hanover.
1827 Published his Disquisitiones Generales Circa Superfices Curvas (General Disquisition on Curved Surfaces), which founded the field of differential geometry of curved surfaces.
1831 Studies crystallograpy.
1831 Publishes his paper on biquadratic residues.
1831 Death of his second wife.
1832 Research on electricity and magnetism.
1833 With Weber, built a 1.4 mile long telegraph linking the observatory to the main university in Göttingen.
1835 Published memoir on magnetism.
1838 First grandchild born, near St. Charles, Missouri.
1838 Begins studying Russian.
1840 Begins studying Sanskrit.
1840 Death of his daughter Minna.
1841 Publishes proof of Legendres's theorem in spherical trigonometry.
1845 Telegraph destroyed by lightning.
1845-1855 Publishes memoirs on geodesy, conducts a variety of astronomical observations, and supervises students, including Reimann.
1855 Gauss dies, February 23^rd.

In addition to all of Gauss's published works, it became clear, long after his death, that he had discovered a great many other mathematical truths which he never published. He kept a tiny notebook, only 19 pages long, that summarized his results, often condensing months of work into a single line. In all, Gauss recorded 146 results in this diary, which was not discovered until 1898, among the posessions of one of his grandsons. Gauss's unpubulished works appear to have included Cauchy's integral theorem, the Taylor expansion, non-euclidean geometry, and the theory of elliptic functions. All ideas that were published by others years after Gauss's discoveries. In some cases, these results were not published until after Gauss's death.

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