Carl Gauss

Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen (help·info), Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

Sometimes referred to as the Princeps mathematicorum^[1] (Latin, "the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity," Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.^[2] He referred to mathematics as "the queen of sciences."^[3]

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 was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.

Contents [hide] 1 Early years (1777–1798) 2 Middle years (1799–1830) 3 Later years and death (1831–1855) 4 Religion 5 Family 6 Personality 7 Commemorations 8 Writings 9 See also 10 Notes 11 Further reading 12 External links

Early years (1777–1798)

Statue of Gauss at his birthplace, Braunschweig

Carl Friedrich Gauss was born on April 30, 1777 in Braunschweig, in the Electorate of Brunswick-Lüneburg, now part of Lower Saxony, Germany, as the son of poor working-class parents.^[4] He was christened and confirmed in a church near the school he attended as a child.^[5] There are several stories of his early genius. According to one, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances.

Another famous story has it that in primary school after the young Gauss misbehaved, his teacher, J.G. Büttner, gave him a task : add a list of integers in arithmetic progression; as the story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant Martin Bartels.

Gauss's presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050. However, the details of the story are at best uncertain (see ^[6] for discussion of the original Wolfgang Sartorius von Waltershausen source and the changes in other versions); some authors, such as Joseph Rotman in his book A first course in Abstract Algebra, question whether it ever happened.

Gauss's intellectual abilities attracted the attention of the Duke of Braunschweig,^[2] who sent him to the Collegium Carolinum (now Technische Universität Braunschweig), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798. While in university, Gauss independently rediscovered several important theorems;^{[citation needed]} his breakthrough occurred in 1796 when he was able to show 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. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.^[7]

The year 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on March 30.^[8] He invented modular arithmetic, greatly simplifying manipulations in number theory.^{[citation needed]} He became the first to prove the quadratic reciprocity law on 8 April. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the famous words, "Heureka! num = Δ + Δ + Δ." On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which ultimately led to the Weil conjectures 150 years later.

Middle years (1799–1830)

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial over the complex numbers has at least one root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to implicit use of the Jordan curve theorem. However, he subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae (Latin, Arithmetical Investigations), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, had the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

Title page of Gauss's Disquisitiones Arithmeticae

In that same year, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres, but could only watch it for a few days. Gauss predicted correctly the position at which it could be found again, and it was rediscovered by Franz Xaver von Zach on 31 December 1801 in Gotha, and one day later by Heinrich Olbers in Bremen.

Gauss's method involved determining a conic section in space, given one focus (the sun) and the conic's intersection with three given lines (lines of sight from the earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work Gauss used comprehensive approximation methods which he created for that purpose.^[9]

Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again." Though Gauss had been up to that point supported by the stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support. Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical observatory in Göttingen, a post he held for the remainder of his life.

The discovery of Ceres by Piazzi on 1 January 1801 led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 under the name Theoria motus corporum coelestium in sectionibus conicis solem ambientum (theory of motion of the celestial bodies moving in conic sections around the sun). Piazzi had only been able to track Ceres for a couple of months, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data—three degrees represent less than 1% of the total orbit.

Gauss, who was 23 at the time, heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree. In the process, he so streamlined the cumbersome mathematics of 18th century orbital prediction that his work—published a few years later as Theory of Celestial Movement—remains a cornerstone of astronomical computation.^{[citation needed]} It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error. Gauss was able to prove the method in 1809 under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1795.^{[citation needed]}

Gauss' portrait published in Astronomische Nachrichten 1828

Gauss was a prodigious mental calculator. Reputedly, when asked how he had been able to predict the trajectory of Ceres with such accuracy he replied, "I used logarithms." The questioner then wanted to know how he had been able to look up so many numbers from the tables so quickly. "Look them up?" Gauss responded. "Who needs to look them up? I just calculate them in my head!"^{[citation needed]}

In 1818 Gauss, putting his calculation skills to practical use, carried out a geodesic survey of the state of Hanover, linking up with previous Danish surveys. To aid in the survey, Gauss invented the heliotrope, an instrument that uses a mirror to reflect sunlight over great distances, to measure positions.

Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."

Four Gaussian distributions in statistics

This unproved statement put a strain on his relationship with János Bolyai (who thought that Gauss was "stealing" his idea), but it is now generally taken at face value.^{[citation needed]} Letters by Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, a life-long student of Gauss, successfully proves in Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidian geometry long before it was published by János Bolyai, but that he refused to publish any of it because of his fear of controversy.

The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces. Among other things he came up with the notion of Gaussian curvature. This led in 1828 to an important theorem, the Theorema Egregium (remarkable theorem in Latin), establishing an important property of the notion of curvature. Informally, the theorem 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 or 2-dimensional space.

In 1821, he was made a foreign member of the Royal Swedish Academy of Sciences.

Later years and death (1831–1855)

Daguerreotype of Gauss on his deathbead, 1855.

Grave of Gauss at Albanifriedhof in Göttingen, Germany.

In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism (including finding a representation for the unit of magnetism in terms of mass, length and time) and the discovery of Kirchhoff's circuit laws in electricity. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. Gauss ordered a magnetic observatory to be built in the garden of the observatory, and with Weber founded the magnetischer Verein (magnetic club in German), which supported measurements of earth's magnetic field in many regions of the world. He developed a method of measuring the horizontal intensity of the magnetic field which has been in use well into the second half of the 20th century and worked out the mathematical theory for separating the inner (core and crust) and outer (magnetospheric) sources of Earth's magnetic field.

Gauss died in Göttingen, Hannover (now part of Lower Saxony, Germany) in 1855 and is interred in the cemetery Albanifriedhof there. Two individuals gave eulogies at his funeral, Gauss's son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer. His brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters^[10] (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of his genius.^[2]

Religion

This section requires expansion.

According to Dunnington, Gauss's religion was based upon the search for truth. He believed in "the immortality of the spiritual individuality, in a personal permanence after death, in a last order of things, in an eternal, righteous, omniscient and omnipotent God." Gauss also upheld religious tolerance, believing it wrong to disturb others who were at peace with their own beliefs.^[2]

Family

Gauss' daughter Therese (1816—1864)

Gauss's personal life was overshadowed by the early death of his first wife, Johanna Osthoff, in 1809, soon followed by the death of one child, Louis. Gauss plunged into a depression from which he never fully recovered. He married again, to Johanna's best friend named Friederica Wilhelmine Waldeck but commonly known as Minna. When his second wife died in 1831 after a long illness,^[11] one of his daughters, Therese, took over the household and cared for Gauss until the end of his life. His mother lived in his house from 1817 until her death in 1839.^[2]

Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). Of all of Gauss's children, Wilhelmina was said to have come closest to his talent, but she died young. With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene emigrated to the United States about 1832 after a falling out with his father.^{[citation needed]} Wilhelm also settled in Missouri, starting as a farmer and later becoming wealthy in the shoe business in St. Louis. Therese kept house for Gauss until his death, after which she married.

Gauss eventually had conflicts with his sons, two of whom migrated to the United States. He did not want any of his sons to enter mathematics or science for "fear of sullying the family name".^{[citation needed]} Gauss wanted Eugene to become a lawyer, but Eugene wanted to study languages. They had an argument over a party Eugene held, which Gauss refused to pay for. The son left in anger and emigrated to the United States, where he was quite successful. It took many years for Eugene's success to counteract his reputation among Gauss's friends and colleagues. See also the letter from Robert Gauss to Felix Klein on 3 September 1912.

Personality

Gauss was an ardent perfectionist and a hard worker. According to Isaac Asimov, Gauss was once interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment till I'm done."^[12] This anecdote is briefly discussed in G. Waldo Dunnington's Gauss, Titan of Science where it is suggested that it is an apocryphal story.

He was never a prolific writer, refusing to publish work which he did not consider complete and above criticism. This was in keeping with his personal motto pauca sed matura ("few, but ripe"). His personal diaries indicate that he had made several important mathematical discoveries years or decades before his contemporaries published them. Mathematical historian Eric Temple Bell estimated that had Gauss timely published all of his discoveries, Gauss would have advanced mathematics by fifty years.^[13]

Though he did take in a few students, Gauss was known to dislike teaching. It is said that he attended only a single scientific conference, which was in Berlin in 1828. However, several of his students became influential mathematicians, among them Richard Dedekind, Bernhard Riemann, and Friedrich Bessel. Before she died, Sophie Germain was recommended by Gauss to receive her honorary degree.

Gauss usually declined to present the intuition behind his often very elegant proofs—he preferred them to appear "out of thin air" and erased all traces of how he discovered them.^{[citation needed]} This is justified, if unsatisfactorily, by Gauss in his "Disquisitiones Arithmeticae", where he states that all analysis (i.e. the paths one travelled to reach the solution of a problem) must be suppressed for sake of brevity.

Gauss supported monarchy and opposed Napoleon, whom he saw as an outgrowth of revolution.

Commemorations

A 10 Deutsche Mark banknote from Germany 1993 (discontinued) showing Gauss

Gauss (about 26) on an East-German stamp produced in 1977. Heptadecagon, compass and straightedge are shown next to him.

From 1989 until the end of 2001, his portrait and a normal distribution curve as well as some prominent buildings of Göttingen were featured on the German ten-mark banknote. The other side of the note features the heliotrope and a triangulation approach for Hannover. Germany has issued three stamps honouring Gauss, as well. A righteous stamp (no. 725), was issued in 1955 on the hundredth anniversary of his death; two other stamps, no. 1246 and 1811, were issued in 1977, the 200th anniversary of his birth.

Daniel Kehlmann's 2005 novel Die Vermessung der Welt, translated into English as Measuring the World: a Novel in 2006, explores Gauss's life and work through a lens of historical fiction, contrasting it with the German explorer Alexander von Humboldt.

In 2007, his bust was introduced to the Walhalla temple.^[14]

Things named in honour of Gauss include:

• The CGS unit for magnetic induction was named gauss in his honour.
• The crater Gauss on the Moon^[15]
• Asteroid 1001 Gaussia.
• The ship Gauss, used in the Gauss expedition to the Antarctic.
• Gaussberg, an extinct volcano discovered by the above mentioned expedition
• Gauss Tower, an observation tower in Dransfeld, Germany.
• In Canadian junior high schools, an annual national mathematics competition administered by the Centre for Education in Mathematics and Computing is named in honour of Gauss.
• In University of California, Santa Cruz, in Crown College, a dormitory building is named after Gauss.
• The Gauss Haus, an NMR center at the University of Utah.
• The Carl-Friedrich-Gauß School for Mathematics, Computer Science, Business Administration, Economics, and Social Sciences of University of Braunschweig.

Writings

• 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse ("New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors [i.e. polynomials] of the first or second degree")

• 1801: Disquisitiones Arithmeticae. German translation by H. Maser Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8 , pp. 1–453. English translation by Arthur A. Clarke Disquisitiones Arithemeticae (Second, corrected edition). New York: Springer. 1986. ISBN 0387962549 .

• 1808: Theorematis arithmetici demonstratio nova. Göttingen: Comment. Soc. regiae sci, Göttingen XVI . German translation by H. Maser Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8 , pp. 457–462 [Introduces Gauss's lemma, uses it in the third proof of quadratic reciprocity]

• 1809: Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium (Theorie der Bewegung der Himmelskörper, die die Sonne in Kegelschnitten umkreisen), English translation by C. H. Davis, reprinted 1963, Dover, New York.

• 1811: Summatio serierun quarundam singularium. Göttingen: Comment. Soc. regiae sci, Göttingen . German translation by H. Maser Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8 , pp. 463–495 [Determination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity]

• 1812: Disquisitiones Generales Circa Seriem Infinitam

• 1818: Theorematis fundamentallis in doctrina de residuis quadraticis demonstrationes et amplicationes novae. Göttingen: Comment. Soc. regiae sci, Göttingen . German translation by H. Maser Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8 , pp. 496–510 [Fifth and sixth proofs of quadratic reciprocity]

• 1821, 1823 und 1826: Theoria combinationis observationum erroribus minimis obnoxiae. Drei Abhandlungen betreffend die Wahrscheinlichkeitsrechnung als Grundlage des Gauß'schen Fehlerfortpflanzungsgesetzes. English translation by G. W. Stewart, 1987, Society for Industrial Mathematics.

• 1827: Disquisitiones generales circa superficies curvas, Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores. Volume VI, pp. 99–146. "General Investigations of Curved Surfaces" (published 1965) Raven Press, New York, translated by A.M.Hiltebeitel and J.C.Morehead.

• 1828: Theoria residuorum biquadraticorum, Commentatio prima. Göttingen: Comment. Soc. regiae sci, Göttingen 6 . German translation by H. Maser Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8 , pp. 511–533 [Elementary facts about biquadratic residues, proves one of the supplements of the law of biquadratic reciprocity (the biquadratic character of 2)]

• 1832: Theoria residuorum biquadraticorum, Commentatio secunda. Göttingen: Comment. Soc. regiae sci, Göttingen 7 . German translation by H. Maser Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition). New York: Chelsea. 1965. ISBN 0-8284-0191-8 , pp. 534–586 [Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i]

• 1843/44: Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Zweiter Band, pp. 3–46

• 1846/47: Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Dritter Band, pp. 3–44

• Mathematisches Tagebuch 1796–1814, Ostwaldts Klassiker, Harri Deutsch Verlag 2005, mit Anmerkungen von Neumamn, ISBN 978-3-8171-3402-1 (English translation with annotations by Jeremy Gray: Expositiones Math. 1984)
• Gauss' collective works are online here This includes German translations of Latin texts and commentaries by various authorities

See also

• Romanticism in science
• German inventors and discoverers
• List of topics named after Carl Friedrich Gauss
• Carl Friedrich Gauss Prize

Notes

1. ^ Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford, UK: Oxford University Press. p. 1188. ISBN 0198507631.
2. ^ ^{a b c d e} Dunnington, G. Waldo. (May, 1927). "The Sesquicentennial of the Birth of Gauss". Scientific Monthly XXIV: 402–414. Retrieved on 29 June 2005. Comprehensive biographical article.
3. ^ Smith, S. A., et al. 2001. Algebra 1: California Edition. Prentice Hall, New Jersey. ISBN 0130442631
4. ^ "Carl Friedrich Gauss". Wichita State University. http://www.math.wichita.edu/history/men/gauss.html.
5. ^ Susan Chambless. "Author — Date". Homepages.rootsweb.ancestry.com. http://homepages.rootsweb.ancestry.com/~schmblss/home/Letters/Gauss/1911-07-26b.htm. Retrieved 2009-07-19.
6. ^ http://www.americanscientist.org/issues/pub/gausss-day-of-reckoning/2
7. ^ Pappas, Theoni: Mathematical Snippets, Page 42. Pgw 2008
8. ^ Carl Friedrich Gauss §§365–366 in Disquisitiones Arithmeticae. Leipzig, Germany, 1801. New Haven, CT: Yale University Press, 1965.
9. ^ Klein, Felix; Hermann, Robert (1979). Development of mathematics in the 19th century. Math Sci Press. ISBN 9780915692286.
10. ^ This reference from 1891 (Donaldson, Henry H. (1891). "Anatomical Observations on the Brain and Several Sense-Organs of the Blind Deaf-Mute, Laura Dewey Bridgman". The American Journal of Psychology (E. C. Sanford) 4 (2): 248–294. doi:10.2307/1411270. http://jstor.org/stable/1411270. ) says: "Gauss, 1492 grm. 957 grm. 219588. sq. mm. ", i.e the unit is square mm. In the later reference: Dunnington (1927), the unit is erroneously reported as square cm, which gives an unreasonably large area, the 1891 reference is more reliable.
11. ^ "Gauss biography". Groups.dcs.st-and.ac.uk. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Gauss.html. Retrieved 2008-09-01.
12. ^ Asimov, I. (1972). Biographical Encyclopedia of Science and Technology; the Lives and Achievements of 1195 Great Scientists from Ancient Times to the Present, Chronologically Arranged.. New York: Doubleday.
13. ^ Bell, E. T. (2009). "Ch. 14: The Prince of Mathematicians: Gauss". Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré. New York: Simon and Schuster. pp. 218–269. ISBN 0-671-46400-0.
14. ^ "Bayerisches Staatsministerium für Wissenschaft, Forschung und Kunst: Startseite". Stmwfk.bayern.de. http://www.stmwfk.bayern.de/downloads/aviso/2004_1_aviso_48-49.pdf. Retrieved 2009-07-19.
15. ^ Andersson, L. E.; Whitaker, E. A., (1982). NASA Catalogue of Lunar Nomenclature. NASA RP-1097.

Further reading

• Dunnington, G. Waldo. (2003). Carl Friedrich Gauss: Titan of Science. The Mathematical Association of America. ISBN 088385547X. OCLC 53933110.
• Gauss, Carl Friedrich (1965). Disquisitiones Arithmeticae. tr. Arthur A. Clarke. Yale University Press. ISBN 0300094736.
• Hall, Tord (1970). Carl Friedrich Gauss: A Biography. Cambridge, MA: MIT Press. ISBN 0262080400. OCLC 185662235.
• Kehlmann, Daniel (2005). Die Vermessung der Welt. Rowohlt. ISBN 3498035282. OCLC 144590801.
• Sartorius von Waltershausen, Wolfgang (1966). Gauss: A Memorial.
• Simmons, J. (1996). The Giant Book of Scientists: The 100 Greatest Minds of All Time. Sydney: The Book Company.
• Tent, Margaret (2006). The Prince of Mathematics: Carl Friedrich Gauss. A K Peters. ISBN 1568814550.

External links

• Carl Friedrich Gauss on PlanetMath
• Complete works
• Gauss and his children
• Gauss biography
• Carl Friedrich Gauss at the Mathematics Genealogy Project
• Carl Friedrich Gauss, Biography at Fermat's Last Theorem Blog.
• Gauss: mathematician of the millennium, by Jürgen Schmidhuber
• English translation of Waltershausen's 1862 biography
• Gauss general website on Gauss
• MNRAS 16 (1856) 80 Obituary
• Carl Friedrich Gauss on the 10 Deutsche Mark banknote
• O'Connor, John J.; Robertson, Edmund F., "Carl Friedrich Gauss", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Gauss.html .
• Carl Friedrich Gauss at Wikiquote

Euclid's Elements

Euclid's Elements (Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises^[1] and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science.

Euclid's Elements has been referred to as the most successful^[2][3] and influential^[4] textbook ever written. Being first set in type in Venice in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press and is estimated to be second only to the Bible in the number of editions published,^[4] with the number reaching well over one thousand.^[5] It was used as the basic text on geometry throughout the Western world for about 2,000 years. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through school books, did it cease to be considered something all educated people had read.

Click here for a very nice presentation and explanation of Euclid's Elements

Did You Know? ... What Does It All Mean?

My college sophomore daughter turned me on to this video yesterday. 11 million hits can't be wrong. Enjoy:

The Prisoner's Dilemma by Carl Sagan

Prisoner's Dilemma
"A New Way To Think About Rules To Live By"

by Carl Sagan, Parade magazine, 28 Nov 1993

Moral codes that seek to regulate human behavior have been with us not only since the dawn of civilization but also among our pre-civilized, and highly social, hunter-gatherer ancestors. And even earlier. Different societies have different codes. Many cultures say one thing and do another. In a few fortunate societies, an inspired lawgiver lays down a set of rules to live by. But many revered codes have failed to establish a long-lived moral order. For example, the codes of Ashoka (India), Hammurabi (Babylon), Lycurgus (Sparta) and Solon (Athens), which once held sway over mighty civilizations, are today largely defunct. Perhaps they misjudged human nature and asked too much of us. Perhaps experience from one epoch or culture is not wholly applicable to another.

In this article, I describe an early effort - tentative but emerging - to approach the matter scientifically.

In our every day lives, as in the momentous affairs of nations, we must decide: What does it mean to do the right thing? How do we deal with an enemy? Should we ever take advantage of someone who treats us kindly? If hurt by a friend, or helped by an enemy, should we reciprocate in kind?

Examples are all around us: Your sister-in-law ignores your snub and invites you over for Christmas dinner. Should you accept? A co-worker makes you look bad in front of the boss. Should you try to get even? Should you cheat at cards? On a larger scale: Should we kill killers? If a power company supports a symphony orchestra, ought we to ignore its destructive, although legal, pollution of the environment? Shattering a worldwide voluntary moratorium, China resumes its testing of nuclear weapons. Should we?

In making such decisions, we're concerned not only with doing right but also with what works - what makes us and the rest of society happier and more secure. There's a tension between what we call ethical and what we call pragmatic. If, even in the long run, ethical behavior were self-defeating, we would not call it ethical, but foolish. (We might even claim to respect it but in practice ignore it.) Bearing in mind the variety and complexity of human behavior, are there any simple rules - whether we call them ethical or pragmatic - that actually work? Let's look at some of the rules we're taught:

THE GOLDEN RULE. The most admired standard of behavior in the West is the Golden Rule. Its formulation in the first-century Gospel of St. Matthew is: "Do unto others as you would have them do unto you." Almost no one follows it consistently. When the Chinese philosopher K'ung-Tzu (known as Confucius in the West) was asked in the sixth century B.C. his opinion of the Golden Rule - of repaying evil with kindness - he replied, "Then with what will you repay kindness?"

THE SILVER RULE. The Silver Rule is different: "Do not do unto others what you would not have them do unto you." The most inspiring 20th-century exemplars of the Silver Rule are Mohandas Gandhi and Dr. Martin Luther King Jr. They counseled oppressed peoples not to repay violence with violence, but not to be compliant and obedient either. Non-violent civil disobedience was what they advocated - putting your body on the line and showing, by your willingness to be punished in defying an unjust law, the justice of your cause. They aimed at melting the hearts of their oppressors. It worked, up to a point. But even Gandhi had trouble reconciling the rule of nonviolence with the necessities of defense against those with less lofty rules of conduct.

THE BRAZEN RULE. "Repay kindness with kindness," said Confucius, describing relations between individuals, "but evil with justice." This might be called the Bronze or Brazen Rule: "Do unto others as they do unto you." It's "an eye for an eye, and a tooth for a tooth," plus "one good turn deserves another." In actual human (and chimpanzee) behavior, it's a familiar standard. Without having to appeal to anyone's better nature, we institute a kind of operant conditioning, rewarding others when they're nice to us and punishing them when they're not. We're not pushovers, be we're not unforgiving either.

THE IRON RULE... AND OTHERS. Of baser coinage is the Iron Rule: "Do unto others as you like, before they do it unto you." It's sometimes formulated as, "He who has the gold makes the rules," underscoring not just its rejection of, but also its contempt for, the Golden Rule. This is the secret maxim of many, if they can get away with it, and often the unspoken precept of the powerful.

Finally, I should mention two mixed rules, found throughout the living world. They explain a great deal. One is: "Suck up to those above you, and intimidate those below." This is the motto of bullies. It's really the Golden Rule for superiors, the Iron Rule for inferiors. Since there is no known alloy of gold and iron, we'll call it the Tin Rule for its flexibility. The other common rule is: "Give precedence in all things to close relatives, and do as you like to others" - the Golden Rule for relatives, the Iron rule for others. This Nepotism Rule is known to evolutionary biologists as "kin selection."

Despite its apparent practicality, there's a fatal flaw in the Brazen Rule: unending vendetta. Each act of justifiable retribution triggers another. Violence begets violence. The reasonable part of us tries to keep the peace, but the passionate part of us cries out for vengeance. Extremists in the two warring factions can count on one another. They are allied against the rest of us, contemptuous of appeals to understanding an loving kindness. A few hotheads can force-march a legion of more prudent and rational people to brutality and war.

WHAT GAMES TEACH US. Clearly, the Brazen Rule is too unforgiving. But the Golden and Silver Rules seem too complacent. They systematically reward cruelty and exploitation. It is hard to imagine a Hitler or a Stalin being shamed into redemption by good example. The Iron Rule promotes the advantage of a ruthless and powerful few against the interest of the many. So is there a rule between the Golden and the Silver, on the one hand, and the Brazen and Iron, on the other, which works better than any of them?

Suppose we seek not to confirm or deny what we've been taught but to find out what really works. Is there a way to test alternative codes of ethics?

We're used to playing games in which somebody wins and somebody loses. Every point made by our opponent puts us that much farther behind. "Win-lose" games seem so natural that many people are hard-pressed to think of a game that isn't win-lose. In win-lose games, the losses just balance the wins - that's why they're also called "zero-sum" games.

Many children are appalled the first time they really come face to face with the "lose" side of win-lose games. On the verge of bankruptcy in the game Monopoly (tm), for example, they plead for special dispensation. When this is not forthcoming, they may, in tears, denounce the game as heartless and unfeeling - which, of course, it is. Within the rules of Monopoly, there's no way for players to cooperate so that all benefit. That's not how the games is designed. The same is true for boxing, football, hockey, basketball, baseball, lacrosse, tennis, racquetball, pinochle, chess, all Olympic events, yacht and car racing, potsy and partisan politics. There may be rewards for teamwork, but not for teamwork with the opponent. In none of these games is there an opportunity to practice the Golden or Silver Rule, or even the Brazen. There is room only for the Rule of Iron.

Nuclear war, however (and many conventional wars), economic depression and assaults on the global environment are all "lose-lose" propositions. Such vital human concerns as love, friendship, parenthood and the pursuit of knowledge are "win-win" propositions. Everyone gains from the creation of great music, art, architecture and literature, wise and just laws and, indeed, far-seeing moral codes. Our vision is dangerously narrow if all we know is "win-lose."

THE PRISONER'S DILEMMA. The scientific field that deals with such matters is called "game theory." It's used in military strategy, trade policy, corporate competition and the limiting of environmental pollution. The Defense Department has its own gaming agency. The paradigmatic game is the Prisoner's Dilemma. It is not zero-sum. Win-win, win-lose and lose-lose outcomes all are possible. It is wholly pragmatic and amoral:

Imagine that you and a friend are arrested for committing a serious crime. Before the two of you have any chance to compare stories or plan strategy, you are taken to separate interrogation cells. There, oblivious of your Miranda rights ("You have the right to remain silent..."), the police try to make you confess. They tell you, as police sometimes do, that your friend has confessed. The police might be telling the truth. Or they might be lying. If you're willing to say anything, what's your best tack to minimize punishment?

You're permitted only to plead guilty or not guilty; you cannot implicate or clear your friend. These are the possible outcomes:

* If you deny committing the crime, and (unknown to you) your friend also denies it, the case might be hard to prove. In the ensuing plea bargain, both your sentences will be very light.

* If you confess, and your friend does likewise, then the effort the State must expend to solve the crime is small. In exchange, you both will be given a fairly light sentence, although not so light as if you both had asserted your innocence.

* If you plead not guilty, and your friend confesses, the State will ask for a maximum sentence for you and minimal punishment (maybe none) for your friend. Uh-oh. You're very vulnerable to a kind of double cross. So's he.

So if you and your friend both plead innocent, you both escape the worst. But each must be sure of the other.

Should you play it safe and guarantee no worse than a middle range of punishment by confessing? Then, if your friend pleads innocent while you plead guilty - well, too bad for him, and you might get off scot-free.

When you think it through, you realize that, whatever your friend does, you're better off confessing. Maddeningly, the same holds true for your friend. But if both of you confess, you both are worse off than if both of you had pleaded innocent. This is the Prisoner's Dilemma.

Robert Axelrod, a professor of political science at the University of Michigan, has pioneered the study of a repeated Prisoner's Dilemma in which the two players go through a sequence of such games with no direct communication between them. At the end of each, they figure out from their punishment how the other must have pleaded. They gain experience about each other's strategy (and character). Will they learn to "cooperate" game after game - both always denying that they committed any crime - even if the reward for finking on the other (or "defecting") is very large?

If you cooperate overmuch, the other player may exploit your good nature. If you defect overmuch, your friend is likely to retaliate often, which will be bad for both of you. What is the right mix of cooperation and defection? How to behave then becomes, like any other question in Nature, a subject to be investigated experimentally.

This matter has been explored by Axelrod in a continuing round-robin computer tournament. Various codes of behavior confront one another, and at the end we see who wins (who gets the lightest cumulative prison term). The simplest strategies might be to cooperate all the time, no matter how much advantage is taken of you; or never to cooperate, no matter what benefits might accrue from cooperation. Both the Golden Rule and the Iron Rule always lose - the one from an excess of kindness, the other from an overabundance of ruthlessness. Strategies that are slow to punish defection lose, in part because they send a signal that non-cooperation works.

A RULE THAT WORKS. The most effective strategy in many such tournaments is called "Tit-for-Tat." It's very simple: You start out cooperating and, in each subsequent round, simply do what your opponent did last time. You punish defections, but once the other player cooperates, you're willing to let bygones be bygones. At first it seems to garner only mediocre success. But as time goes on, the other strategies defeat themselves - from too much kindness or too much cruelty - and this middle way pulls ahead.

Except for always being nice on the first move, Tit-for-tat is identical to the Brazen Rule. It promptly (in the very next game) rewards cooperation and punishes defection, and it has the great virtue that it makes its strategy absolutely clear.

To succeed, Tit-for-tat strategists must find others who are willing to reciprocate - players with whom to cooperate. Once there get to be several players employing Tit-for-tat, they rise in the standings together. After the first tournament, in which the Brazen Rule unexpectedly won, some experts thought it would pay to be less forgiving. Next tournament, they tried to exploit the Brazen Rule by defecting more often. They did poorly. Even experienced strategists tended to underestimate the power of forgiveness and reconciliation.

The superiority of the Brazen Rule in such tournaments was discovered by Axelrod and described in his remarkable book "The Evolution of Cooperation." A variant of Tit-for-tat that forgives other players for defecting occasionally - say 10 percent of the time - does even better if there's any chance of misunderstanding. We might call it the Goldplated Brazen Rule. Among other virtues, it breaks out of unending vendetta.

The Prisoner's Dilemma is a very simple game. Real life is considerably more complex. But its central lessons are striking: Be friendly at first meetings. Do not envy. Be generous; forgive your enemy if he forgives you. Be neither a tyrant or a patsy. Retaliate proportionately to an intentional injury (within the constraints of the rule of law). And make your behavior fairly (although not perfectly) clear and consistent. What would the world be like if more of us, individuals as well as nations, lived by these rules?

America Loses Her Edge In Space + Humanoid Robot Prototype

U.S. Astronaut John Young (yes, he's aged some) cozies up to the Japanese Super Roomba 1, the first humanoid robot prototype, to be delivered to the ISS Nov. 1 aboard the U.S. Space Shuttle

The Space Bill passed Congress, with US Manned Spaceflight being kept alive a bit longer, as explained here.

Some notes:

1) The Commercial funding is 1.3 Billion dollars US, not the 3.3 that Barry Obama requested. Either Elon Musk and Richard Branson were asking for triple the amount they felt they needed, or they should be pretty unhappy.

2) " But the extra space mission would not affect the coming Oct. 1 layoffs of nearly 1,400 shuttle workers by NASA contractor United Space Alliance – a joint venture by Boeing and Lockheed Martin that oversees NASA's shuttle fleet. USA announced the shuttle worker layoffs in July as part of a workforce reduction plan due to the space shuttle fleet's impending retirement. "

Ah, so Boeing and Arms Merchant Extraordinaire Lockheed-Martin were involved, eh? Well there you have it then. Big Money = Bill Passage. That's the Conservation of Economical and Political Money/Energy Law (a.k.a. The Big Money Baksheesh Lobbyist Certainty Principle) from Political "Science", in action for you.

3) "President Obama's new space plan, announced in February, cancelled NASA's moon-oriented Constellation program set forth by former President George W. Bush and called for more ambitious deep space missions to an asteroid and Mars. The Constellation program was responsible for the Orion space capsules and Ares rockets set to follow the shuttle program."

Lord knows I have few kind words for Dubya, but come ON, people! Is there a single astrophysicist with a pair of BALLS large enough to stand up to these folks ?! Take a break from perturbating behind the closed doors of your offices, willya, just for a minute, and consider ...

Our priorities (USA) in manned space exploration seem to be:

T1) Mars
T1) Near-Earth Asteroid
3) Paying off the Russians to ferry our Astroboys'n'girls to the ISS, if they'd be so kind

When in fact they should be:

1) Moon
2) Mercury
3) The Asteroid Belt
4) Mars

or even:

T1) Moon Caves
T1) Moon Poles
T1) Moon Farside Space Observatory
T1) Low Moon Orbit Satellite Communications Network
T1) Moon Spaceport for Solar System Exploration (1/4 earth's gravity doesn't suck. Well it does because gravity sucks, but only 1/4 as much as deep gravity well Earth).
6) Everything else

Once again, I strongly call for Human settlement of our double planet twin. I believe it will happen, but at this rate all the signs on the moon will be in Chinese. America's bugged out, dammit!

- Steve Colyer

P.S. Also from the article:

Most interesting to me was the following passage:

"NASA and its contractors are currently preparing the shuttle Discovery to launch Nov. 1 to deliver a new storage room and humanoid robot prototype to the station."

Human prototype robot ... what?? No, that's not a pic of John Young and the robot above in what passes for my sad sense of humor, it's a picture of Dr. Zachary Smith and "Robot" from the cheesy schlocky 1960's American "sci-fi" TV series, "Lost in Space."

Still ... "Human prototype robot"?! Can she whistle Dixie, and does she eat crackers in bed? Both at the same time? Just another sad thought, sorry.

UPDATE (Oct. 2, 2010): Lori Garver, the Number 2 person at NASA, defends herself, rather badly in my opinion, here. But don't believe me, read the replies and you'll see. Sheesh.