Carl Friedrich Gauss (1777-1855) was one of the greatest mathematicians of all time. He combined scientific theory and practice like no other before him, or since, and even as a young man Gauss made extraordinary contributions to mathematics. His Disquisitiones arithmeticae, published in 1801, stands to this day as a true masterpiece of scientific investigation. In the same year, Gauss gained fame in wider circles for his prediction, using very few observations, of when and where the asteroid Ceres would next appear. The method of least squares, developed by Gauss as an aid in his mapping of the state of Hannover, is still an indispensable tool for analyzing data. His sextant is pictured on the last series of German 10-Mark notes, honoring his considerable contributions to surveying. There, one also finds a bell curve, which is the graphical representation of the Gaussian normal distribution in probability. Together with Wilhelm Weber, Gauss invented the first electric telegraph. In recognition of his contributions to the theory of electromagnetism, the international unit of magnetic induction is the gauss. Left picture and caption taken from Gauss Prize site. Right picture of former German currency taken from Jacob Lewis Bourjaily's home page.

1777-1855

## Biographies of Gauss

• Wikipedia article
"Gauss was a child prodigy. There are many anecdotes pertaining to his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his magnum opus, in 1798 at the age of 21, though it would not be published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day."

• MathWorld article
"German mathematician who is sometimes called the 'prince of mathematics.' He was a prodigious child, at the age of three informing his father of an arithmetical error in a complicated payroll calculation and stating the correct answer. In school, when his teacher gave the problem of summing the integers from 1 to 100 (an arithmetic series) to his students to keep them busy, Gauss immediately wrote down the correct answer 5050 on his slate. At age 19, Gauss demonstrated a method for constructing a heptadecagon using only a straightedge and compass which had eluded the Greeks."

• St. Andrews biography
"At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels, were 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 101."

• Geocities biography Includes a list of formulas discovered by Gauss
"From the outside, Gauss' life was very simple. Having brought up in an austere childhood in a poor and uneducated family he showed extraordinary precocity. He received a stipend from the duke of Brunswick starting at the age of 14 which allowed him to devote his time to his studies for 16 years. Before his 25th birthday, he was already famous for his work in mathematics and astronomy. When he became 30 he went to Göttingen to become director of the observatory. He rarely left the city except on scientific business. From there, he worked for 47 years until his death at almost 78. In contrast to his external simplicity, Gauss' personal life was tragic and complicated. Due to the French Revolution, Napoleonic period and the democratic revolutions in Germany, he suffered from political turmoil and financial insecurity. He found no fellow mathematical collaborators and worked alone for most of his life. An unsympathetic father, the early death of his first wife, the poor health of his second wife, and terrible relations with his sons denied him a family sanctuary until late in life."
"Even with all of these troubles, Gauss kept an amazingly rich scientific activity. An early passion for numbers and calculations extended first to the theory of numbers, to algebra, analysis, geometry, probability, and the theory of errors. At the same time, he carried on intensive empirical and theoretical research in many branches of science, including observational astronomy, celestial mechanics, surveying, geodesy, capillarity, geomagnetism, electromagnetism, mechanism optics, actuarial science. His publications, abundant correspondence, notes, and manuscripts show him to have been one of the greatest scientific virtuosos of all time."

• Carl Friedrich Gauss: Titan of Science by G. Waldo Dunnington, Jeremy Gray and Fritz-Egbert Dohse

• Gauss: A Biographical Study by W. K. Bühler

## The law of quadratic recipocity, Gauss' "Golden Theorem"

• Wikipedia article "The law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which gives conditions for the solvability of quadratic equations modulo prime numbers."

• "Proofs of quadratic reciprocity In the mathematical field of number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusual number of proofs. Several hundred proofs of the law of quadratic reciprocity have been found."

• MathWorld article "Also called the aureum theorema (golden theorem) by Gauss."

• Gauß, Eisenstein, and the third'' proof of the Quadratic Reciprocity Theorem: Ein kleines Schauspiel
A short play written by my friend and former student David J. Pengelley and Reinhard C. Laubenbacher in 1994.

• A brief summary from the book NUMBER THEORY WITH COMPUTER APPLICATIONS by Ramanujachary Kumanduri and Cristina Romero

"The elementary properties of quadratic congruences and a method for their solution were studied in a previous chapter. Now, we focus our attention on some deeper properties of numbers that were discovered by Euler, Legendre, and Gauss. The simplest of these are the following.

1. The odd prime divisors of numbers of the form are of the form 4k+1.
2. The odd prime divisors of numbers of the form are of the form 8k+1 or 8k-1.
3. The odd prime divisors (also not 3) of numbers of the form are of the form 12k+1 or 12k-1.
4. The prime divisors (not equal to 2 or 5) of numbers of the form are of the form 20k+1, 20k-1, 20k+9 or 20k-9.

Generalizing these results, Euler conjectured that the prime divisors p of numbers of the form are of the form or , for some odd b. This is the Quadratic Reciprocity Law. The first complete proof of this law was given by Gauss in 1796. Gauss gave eight different proofs of the law and we discuss a proof that Gauss gave in 1808."

## The heptadecagon (17-sided polygon), Gauss' first mathematical triumph

• Compass and straightedge - the regular Heptadecagon YouTube video (1:39) showing the ruler and compass construction set to music. Unfortunately, the aspect ratio is wrong, making the circles look like ellipses. The YouTube site has a written explanation of the construction, which is not easy to follow otherwise.

• All the possible polygons! YouTube video (4:48) showing ruler and compass constructions of all possible n-gons for n ≤ 51.

• Wikipedia article on Ruler and compass constructions Includes Gauss' formula

$\cos{\left(\frac{2\pi}{17}\right)} = -\frac{1}{16} \; + \; \frac{1}{16} \sqrt{17} \;+\; \frac{1}{16} \sqrt{34 - 2 \sqrt{17}} \;+\; \frac{1}{8} \sqrt{ 17 + 3 \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{34 + 2 \sqrt{17}} }$

• Construction of the 17-gon
"Although mathematicians have been working on this problem in Euclid's time (300 B.C.E.), it was Gauss who discovered it's construction in 1796 when he was an eighteen-year-old. Another significance is it is by this discovery that Gauss decided to spend him life persuing mathematics."

• More detailed Wikipedia article
"Carl Friedrich Gauss proved - as a 19 year old student at Göttingen University - that the regular heptadecagon (a 17 sided polygon) is constructible with a pair of compasses and a straightedge."

• Another Wikipedia article with graphic animations of the construction and that of the pentagon.

• MathWorld article
"Gauss's proof appears in his monumental work Disquisitiones Arithmeticae. The proof relies on the property of irreducible polynomial equations that roots composed of a finite number of square root extractions only exist when the order of the equation is a product of the form... "

## Finding the orbit of the asteroid Ceres, discovered on the night of January 1, 1801. This made Gauss a celebrity in Europe.

• How Gauss Determined the Orbit of CeresVeronique Le Corvec, Jeffrey Donatelli and Jeffrey Hunt, 15 page slide presntation.

• The Discovery of Ceres: How Gauss Became Famous by Donald Teets and Karen Whitehead. 11 page article from Mathematics Magazine, Vol. 72, No. 2 (Apr., 1999): pp. 83-93.

• The Discovery of Ceres Short online article with moving illustrations of the geometry involved from Kepler's Discovery .
"Gauss discovered a method for computing the planet's orbit using only three of the original observations and successfully predicted where Ceres might be found."
"The prediction catapulted him to worldwide acclaim, due, in the words of biographer W. K. Bühler, 'to the popular appeal which astronomy has always enjoyed,' and launched one of the most fruitful careers in the history of science."

• Gauss and Ceres by Leorah Weiss. A short account written by a student at Rutgers.
"... for it is now clearly shown that the orbit of a heavenly body may be determined quite nearly from good observations embracing only a few days; and this without any hypothetical assumption."
- Carl Friedrich Gauss

• Gauss's Orbits Short article by Ivars Peterson
"With a major mathematical work just published and little else to occupy his time during the latter part of 1801, Gauss brought his formidable powers to bear on celestial mechanics. Like a skillful mechanic, he systematically disassembled the creaky, ponderous engine that had long been used for determining approximate orbits and rebuilt it into an efficient, streamlined machine that could function reliably given even minimal data."

## Gauss and the prime number theorem

• How Many Primes Are There? A short introduction to the problem
"Gauss was also studying prime tables and came up with a different estimate (perhaps first considered in 1791), communicated in a letter to Encke in 1849 and first published in 1863.
Gauss:
pi(x) is approximately Li(x) (the principal value of integral of 1/log u from u=0 to u=x)."
• MathWord article
"In 1792, when only 15 years old, Gauss proposed that ..."

• Wikipedia article
"Carl Friedrich Gauss considered the same question and, based on the computational evidence available to him and on some heuristic reasoning, he came up with his own approximating function, the logarithmic integral li(x), although he did not publish his results."

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