Johann Carl Friedrich Gauss

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Johann Carl Friedrich Gauss

Carl Friedrich Gauss.jpg
Johann Carl Friedrich Gauss,
painted by Christian Albrecht Jensen
Born

April 30, 1777
Brunswick, Germany

Died February 23, 1855

Göttingen, Hannover, Germany

Residence Flag of Germany.svg Germany
Nationality Flag of Germany.svg German
Field Mathematician and physicist
Institutions Georg-August University
Alma mater Helmstedt University
Academic advisor  Johann Friedrich Pfaff
Notable students  Friedrich Bessel

Christoph Gudermann
Christian Ludwig Gerling
J. W. Richard Dedekind
Johann Encke
Johann Listing
Bernhard Riemann

Known for Number theory
The Gaussian
Magnetism

Johann Carl Friedrich Gauss (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. He is particularly known for the unit of magnetism that bears his name, and by a mathematical expression (Gauss's Law) that defines the character of a number of forces and physical phenomena such as electricity, magnetism, gravitation and heat flow.

Contents

Gauss was a deeply religious man with strong convictions, but was tolerant of those with other views. His spiritual intuitions sprung from his love of truth and righteousness. He believed in a life beyond the grave.

Biography

Statue of Gauss in Brunswick

Gauss was born in Brunswick, in the Duchy of Brunswick-Lüneburg (now part of Lower Saxony, Germany), as the only son of working-class parents. According to legend, his gifts became very apparent at the age of three when he corrected, in his head, an error his father had made on paper while calculating finances.

Early years

In 1884, at age 7, he entered public elementary school. A famous story, and one that has evolved in the telling, has it that his primary school teacher, J.G. Büttner tried to occupy pupils by making them add up the integers from 1 to 100. The young Gauss produced the correct answer within seconds by a flash of mathematical insight, to the astonishment of all. Gauss had realized 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 (see arithmetic series and summation).[1] At the age of 10, he befriended a teacher's assistant who helped Gauss procure books on mathematics, which they studied together. Gauss began to attract the attention of influential people in the court of Karl Wilhelm Ferdinand, Duke of Brunswick-Luneburg. In 1888, he was admitted to gymnasium (high school), but after two years, having excelled to a remarkable degree in his studies, he was presented to the duke, who awarded Gauss a fellowship to the Collegium Carolinum (now Technische Universität Braunschweig), which he attended from 1792 to 1795. From there Gauss went on to the University of Göttingen from 1795 to 1798.

Mathematical discoveries

While in college, Gauss independently rediscovered several important theorems; his breakthrough occurred in 1796 when he was able to show that a 17-sided polygon 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. 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. At the same time, he discovered the method of least squares, used to estimate the impact of random errors in measurement.

The year 1796 was probably the most productive for both Gauss and number theory. He invented modular arithmetic, a field dealing with number systems that repeat, such as in 24-hour clock time. He greatly simplified manipulations in number theory. He became the first to prove the quadratic reciprocity law on April 8. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on May 31, gives a good understanding of how the prime numbers (odd integers that are not divisible by any other integer except 1) are distributed among the integers. On October 1, he published a result on the theory of solutions of polynomial equations.

Middle years

Title page of Gauss's Disquisitiones Arithmeticae

In his 1799 dissertation, A New Proof That Every Rational Integer Function of One Variable Can Be Resolved into Real Factors of the First or Second Degree, Gauss gave a proof of the fundamental theorem of algebra. This important theorem states that every polynomial over the complex numbers must have at least one root. Other mathematicians had tried to prove this before him, e.g. Jean le Rond d'Alembert. Gauss's dissertation contained a critique of d'Alembert's proof, but his own attempt would not be accepted owing to implicit use of the Jordan curve theorem which deals with boundaries between closed loops and the surfaces that contain them. Gauss over his lifetime produced three more proofs, probably due in part to this rejection of his dissertation; his last proof in 1849 is generally considered rigorous by today's standard. 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, which contained a clean presentation of modular arithmetic and the first proof of the law of quadratic reciprocity. Also in 1801, He was elected as a corresponding member of the St. Petersburg Academy of Science.

Astronomical researches

In that same year, Italian astronomer Giuseppe Piazzi discovered the planetoid Ceres, but could only watch it for a few days before it disappeared in the glare of the sun. Gauss, who was 23 at the time, heard about the problem and tackled it head-on. 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. It was rediscovered by Franz Xaver von Zach on December 31, 1801 in Gotha, and one day later by Heinrich Olbers in Bremen. Zach noted that "Ceres is now easy to find and can never again be lost, since the ellipse of Dr. Gauss agrees so exactly with its location." Other astronomers working on the same problem had failed to establish an accurate orbit for Ceres.

In 1802, Gauss applied his talents to establishing the orbit of another then-recently discovered asteroid, Pallas. For this work, he was awarded a medal by the Institute of France in 1810 for the most original astronomical observation.

Though Gauss had up to this point been supported by the stipend from the Duke Wilhelm (who would suffer a fatal wound during a battle with Napoleon's army in 1806), 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.

Gauss's first years at Gottingen were troubled by the deaths of his father in 1807, and his wife in 1809. He was also subject to a heavy tax imposed by Napoleon, which made him liable for two thousand francs. The famed mathematical physicist Pierre-Simon Laplace paid this for him, but Gauss, who felt uncomfortable with the unsolicited gesture, returned the amount to Laplace with interest for the time elapsed.

Gauss's work on on the orbit of Ceres led to his development of 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).

In the process of his investigation, he so streamlined the cumbersome mathematics of eighteenth century orbital prediction that his work remains a cornerstone of astronomical computation. 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 demonstrate the rigor of 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.

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!"

Surveying

Gauss had been asked in the late 1810s to carry out a geodetic survey of the state of Hanover to link up with the existing Danish grid. Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations. He regularly wrote to Schumacher, Olbers and Bessel, reporting on his progress and discussing problems. As part of the survey, Gauss invented the heliotrope. This instrument produces a bright light by reflecting the Sun's rays using a set of mirrors and a small telescope, so that positions can be accurately determined from afar by surveyors.

Non-Euclidean Geometry

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." This unproved statement put a strain on his relationship with János Bolyai (who thought that Gauss was "stealing" his idea), but it is nowadays generally taken at face value.

Four Gaussian distributions in statistics.

The survey of Hanover later led to the development of the Gaussian distribution, also known as the normal distribution, for describing measurement errors. Moreover, it fuelled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces. In this field, he came up in 1828 with an important theorem, the theorema egregrium (remarkable theorem in Latin) establishing an important property of the notion of curvature.

Later years, death, and afterwards

In 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber; it led to new knowledge in the field of 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. Gauss and Weber constructed the first electromagnetic 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"), 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 twentieth century and worked out the mathematical theory for separating the inner (core and crust) and outer (magnetospheric) sources of Earth's magnetic field.

Gauss customarily avoided anything having to do with the medical profession, but later in life, at the urging of friends, he sought the advice of a physician. Having suffered from shortness of breath and congestion in his last years, he was diagnosed as having an enlarged heart. Gauss died in Göttingen, Hanover (now part of Lower Saxony, Germany), from what was most likely congestive heart failure, on Febuary 23, 1855. He 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 weight to be 1,492 grams and the cerebral area equal to 219,588 square centimeters. Highly developed convolutions were also found, which in the early twentieth century was suggested as the explanation of his genius (Dunnington 1927).

Other notable contributions and associations

Telegraph

In 1834, Gauss, with the help of Weber, set up a telegraph line between two stations within the campus of their magnetic observatory in Gottingen, and were able to send and receive messages. This represents one of the earliest systems of electronic telegraphy. The Gauss/Weber system was capable of sending about 8 words a minute. In 1836, a plan was developed for a telegraphic link between Leipzig and Dresden based on the Gauss/Weber device. The plan was scrapped when the railroad sponsoring the venture ran into financial difficulties.

Gauss's law

Gauss's Law is a simple way to describe the relationship between force fields or other phenomena that follow the inverse square law. Gravitation, magnetism and static electricity obey this law. It can only be expressed in the complex language of infinitesimal calculus.

When applied to heat transfer, it is equivalent to saying that the net flow of heat out of a closed surface such as a sphere or cylinder is proportional to the rate at which heat is supplied by the sources in the volume contained by the surface.

Gaussian distribution

Also referred to as standard distribution, the gaussian distribution is applied to random errors of measurement, and is sometimes referred to as a bell curve because of its shape when represented graphically. It is used to determine the most likely value of a parameter from a number of measurements that follow a statistical pattern of error. Gauss used it to process data on astronomical positions.

Magnetic flux intensity

The unit of magnetic flux intensity is the gauss, and is defined as one Maxwell per square centimeter. As a unit, it is represented by the letter G, although the magnetic flux intensity itself is generally designated by the letter B in equations.

Family

Gauss was married twice. He married his first wife, Johanna Osthoff, in 1805. Johanna died in 1809, and Louis died soon afterward. Gauss plunged into a depression from which he never fully recovered. He married again, to a friend of his first wife named Friederica Wilhelmine Waldeck (Minna), but this second marriage does not seem to have been very happy. When his second wife died in 1831 after a long illness, 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.

Gauss had six children, three by each wife. 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 immigrated to the United States about 1832 after a falling out with his father, eventually settling in St. Charles, Missouri, where he became a well respected member of the community. Wilhelm came to settle in Missouri somewhat later, 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.

Personality

Gauss was an ardent perfectionist and a hard worker. He was never a prolific writer, refusing to publish works which he did not consider complete and above criticism. This was in keeping with his personal motto pauca sed matura (few, but ripe). A study of his personal diaries reveals that he had in fact discovered several important mathematical concepts years or decades before they were published by his contemporaries. Prominent mathematical historian Eric Temple Bell estimated that had Gauss made known all of his discoveries, mathematics would have been advanced by fifty years. (Bell, 1937)

Another criticism of Gauss is that he did not support the younger mathematicians who followed him. He rarely, if ever, collaborated with other mathematicians and was considered aloof and austere by many. 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 turned out to be 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's father was not supportive of Gauss's schooling, and he was primarily supported by his mother in this effort. Likewise, he 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." His conflict with Eugene was particularly bitter. 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 immigrated 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 September 3, 1912.

Unlike modern mathematicians, 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.

Gauss was deeply religious and conservative. He supported monarchy and opposed Napoleon whom he saw as an outgrowth of revolution.

His biographer and close associate during his lifetime, W. S. Von Waltershausen, wrote: ..."the search for truth and the feeling for righteousness were the basis of his religious views. Thus he conceived of spiritual life through the universe as a state of righteousness penetrated by eternal truth. From this he drew the trust, the confidence that our life course is not ended by death."

Commemorations

The cgs unit for magnetic induction was named gauss in his honor.

From 1989 until the end of 2001, his portrait and a normal distribution curve were featured on the German ten-mark banknote. Germany has issued three stamps honoring Gauss, as well. A 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 two-hundreth anniversary of his birth.

G. Waldo Dunnington was a lifelong student of Gauss. He wrote many articles, and a biography: Carl Frederick Gauss: Titan of Science. This book was reissued in 2003, after having been out of print for almost 50 years.

In 2007, his bust will be introduced to the Walhalla.

Places, vessels and events named in honour of Gauss:

  • Gauss crater on the Moon
  • Asteroid 1001 Gaussia.
  • The First German Antarctica Expedition's ship Gauss
  • Gaussberg, an extinct volcano discovered by the above mentioned expedition
  • Gauss Tower, an observation tower
  • 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.

See also

  • Gaussian
  • Magnetism
  • Mathematics
  • Number theory

Notes

  1. Gauss's Day of Reckoning, Sigma Xi, The Scientific Research Society. Retrieved March 31, 2008.

References

  • Bell, E. T. 1986. Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré. New York: Simon and Schuster. 218–269. ISBN 067146400
  • Dunnington, G. Waldo. 2003. Carl Friedrich Gauss: Titan of Science. The Mathematical Association of America. ISBN 088385547X
  • Gauss, Carl Friedrich. 1965. Disquisitiones Arithmeticae translated by Arthur A. Clarke. New Haven: Yale University Press. ISBN 0300094736
  • Gillispie, Charles Coulston. 1975. Dictionary of Scientific Biography. New York: Scribner. ISBN 0684101211.
  • Hall, T. 1970. Carl Friedrich Gauss: A Biography. Cambridge, MA: MIT Press. ASIN B000IOVP7A. ISBN 0262080400l.
  • Simmons, J. 1996. The Giant Book of Scientists: The 100 Greatest Minds of All Time. Sydney: The Book Company.
  • Von Waltershausen, W. S. 1856. Carl Friederich Gauss: A Memorial. Leipzig: H. Hirzel.
  • 1856. Proceedings of the Royal Society of London. London: Taylor and Francis. 7:589-598.

External links

All links retrieved December 6, 2013.

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