Johann Carl Friedrich Gauss

Question

I've been asked to give a brief over view of the mathematician Carl Gauss' life, so i should include his birth, family, education, occupation, death, who influenced him, his major contributions and theorems.

I understand that this topic is extremely vast and as a mathematician he is considered as one of the greatest. I was wondering if anyone has a useless link which summaries a lot of this information.

Answer 1

This is from "Mathematical Physics, A Modern Introduction to Its Foundations", by Sadri Hassani, p. 416: (Every time I read this my hair stands up! What a giant Gauss was indeed...)

Johann Carl Friedrich Gauss (1777-1855)was the greatest of all mathematicians and perhaps the most richly gifted genius of whom there is any record. He was born in the city of Brunswick in northern Germany, His exceptional skill with numbers was clear at a very early age, and in later life he joked that he knew how to count before he could talk. It is said that Goethe wrote and directed little plays for a puppet theater when he was 6 and that Mozart composed his first childish minuets when he was 5, but Gauss corrected an error in his father's payroll accounts at the age of 3. At the age of seven, when he started elementary school, his teacher was amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 1O1.

His long professional life is so filled with accomplishments that it is impossible to give a full account of them in the short space available here. All we can do is simply give a chronology of his almost uncountable discoveries.

1792-1794: Gauss reads the works of Newton, Euler, and Lagrange; discovers the prime number theorem (at the age of 14 or 15); invents the method of least squares; conceives the Gaussian law of distribution in the theory of probability.

1795: (only 18 years old!) Proves that a regular polygon with $n$ sides is constructible (by ruler and compass) if and only if $n$ is the product of a power of 2 and distinct prime numbers of the form $p_k = 2^{2k+1}$, and completely solves the 2000-year old problem of ruler-and-compass construction of regular polygons. He also discovers the law of quadratic reciprocity.

1799: Proves the fundamental theorem of algebra in his doctoral dissertation using the then-mysterious complex numbers with complete confidence.

1801: Gauss publishes his Disquisitiones Arithmeticae in which he creates the modem rigorous approach to mathematics; predicts the exact location of the asteroid Ceres.

1807: Becomes professor of astronomy and the director of the new observatory at Gottingen.

1809: Publishes his second book, Theoria motus corporum coelestium, a major two-volume treatise on the motion of celestial bodies and the bible of planetary astronomers for the next 100years.

1812: Publishes Disquisitiones generales circa seriem infinitam, a rigorous treatment of infinite series, and introduces the hypergeometric function for the first time, for which he uses the notation $F(\alpha, \beta, \gamma;z)$; an essay on approximate integration.

1820-1830: Publishes over 70 papers, including Disquisitiones generales circa superficies curvas, in which he creates the intrinsic differential geometry of general curved surfaces, the forerunner of Riemannian geometry and the general theory of relativity. From the 1830s on, Gauss was increasingly occupied with physics, and he enriched every branch of the subject he touched. In the theory of surface tension, he developed the fundamental idea of conservation of energy and solved the earliest problem in the calculus of variations. In optics, he introduced the concept of the focal length of a system of lenses. He virtually created the science of geomagnetism, and in collaboration with his friend and colleague Wilhelm Weber he invented the electromagnetic telegraph. In 1839 Gauss published his fundamental paper on the general theory of inverse square forces, which established potential theory as a coherent branch of mathematics and in which he established the divergence theorem.

Gauss had many opportunities to leave Gottingen, but he refused all offers and remained there for the rest of his life, living quietly and simply, traveling rarely, and working with immense energy on a wide variety of problems in mathematics and its applications. Apart from science and his family-he married twice and had six children, two of whom emigrated to America-his main interests were history and world literature, international politics, and public finance. He owned a large library of about 6000 volumes in many languages, including Greek, Latin, English, French, Russian, Danish, and of course German. His acuteness in handling his own financial affairs is shown by the fact that although he started with virtually nothing, he left an estate over a hundred times as great as his average annual income during the last half of his life. The foregoing list is the published portion of Gauss's total achievement; the unpublished and private part is almost equally impressive. His scientific diary, a little booklet of 19 pages, discovered in 1898, extends from 1796 to 1814 and consists of 146 very concise statements of the results of his investigations, which often occupied him for weeks or months. These ideas were so abundant and so frequent that he physically did not have time to publish them. Some of the ideas recorded in this diary:

Cauchy Integral Formula: Gauss discovers it in 1811, 16 years before Cauchy.

Non-Euclidean Geometry: After failing to prove Euclid's fifth postulate at the age of 15, Gauss came to the conclusion that the Euclidean form of geometry cannot be the only one possible.

Elliptic Functions: Gauss had found many of the results of Abel and Jacobi (the two main contributors to the subject) before these men were born. The facts became known partly through Jacobi himself. His attention was caught by a cryptic passage in the Disquisitiones, whose meaning can only be understood if one knows something about elliptic functions. He visited Gauss on several occasions to verify his suspicions and tell him about his own most recent discoveries, and each time Gauss pulled 30-year-old manuscripts out ofhis desk and showed Jacobi what Jacobi had just shown him. After a week's visit with Gauss in 1840, Jacobi wrote to his brother, "Mathematics would be in a very different position if practical astronomy had not diverted this colossal genius from his glorious career."

A possible explanation for not publishing such important ideas is suggested by his comments in a letter to Bolyai: "It is not knowledge but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it in order to go into darkness again." His was the temperament of an explorer who is reluctant to take the time to write an account of his last expedition when he could be starting another. As it was, Gauss wrote a great deal, but to have published every fundamental discovery he made in a form satisfactory to himself would have required several long lifetimes.