Everything about Carl Friedrich Gauss totally explained
Johann Carl Friedrich 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,
electrostatics,
astronomy, and
optics. Sometimes known as the
princeps mathematicorum (
Latin, usually translated as "the Prince of Mathematicians", although Latin
princeps also can simply mean "the foremost") 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.
Gauss was a
child prodigy, of whom there are many
anecdotes pertaining to his astounding precocity while a mere toddler, and 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 wouldn't be published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
Early years (1777–1798)
Gauss was born in
Braunschweig, in the
Electorate of Brunswick-Lüneburg, now part of
Lower Saxony,
Germany, as the only son of poor working-class parents. There are several stories of his early genius, all of them open to doubt; according to one, 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.
Another famous story, and one that has evolved in the telling, has it that in
primary school his teacher, J.G. Büttner, tried to occupy pupils by making them add a list of
integers. The young Gauss reputedly produced the correct answer within seconds, to the astonishment of his teacher and his assistant
Martin Bartels. Gauss' presumed method, which supposes the list of numbers was from 1 to 100, was to realise 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). However whilst the method works, the incident itself is probably
apocryphal; some, such as Joseph Rotman in his book
A first course in Abstract Algebra, question whether it ever happened.
His father had wanted him to follow in his footsteps and become a
mason. He wasn't supportive of Gauss's schooling in mathematics and science. Gauss was primarily supported by his mother in this effort and by the
Duke of Braunschweig, He invented
modular arithmetic, greatly simplifying 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 are distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of at most three
triangular numbers on
July 10 and then jotted down in his diary the famous words, "
Heureka! num =
." 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 isn't acceptable, owing to implicit use of the
Jordan curve theorem. However, he subsequently produced three other proofs, the last one in 1849 being generally considered 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 contained a clean presentation of
modular arithmetic and the first proof of the law of
quadratic reciprocity. 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
December 31,
1801 in
Gotha, and one day later by
Heinrich Olbers in
Bremen. 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 didn't 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
January 1,
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 couldn't 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. 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'd been using it since 1795.
Gauss was a prodigious
mental calculator. Reputedly, when asked how he'd been able to predict the trajectory of Ceres with such accuracy he replied, "I used
logarithms." The questioner then wanted to know how he'd 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!"
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." This unproved statement put a strain on his relationship with János Bolyai (who thought that Gauss was "stealing" his idea), but it's now generally taken at face value. 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, 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. 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 doesn't depend on how the surface might be
embedded in 3-dimensional space.
Later years and death (1831–1855)
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
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 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,
Hanover (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 weight to be 1,492 grams and the cerebral area equal to 219,588 square centimeters (236.363 square feet). Highly developed convolutions were also found, which in the early 20th century was suggested as the explanation of his genius.
Family
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. This second marriage doesn't seem to have been very happy as it was plagued by Minna's continuous illness. 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. This anecdote is briefly discussed in Waldo Dunnington's
Gauss, Titan of Science where it's suggested that it's an apocryphal story.
He was never a prolific writer, refusing to publish works which he didn't 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'd in fact discovered several important mathematical concepts years or decades before they were published by his contemporaries. Mathematical historian
Eric Temple Bell estimated that had Gauss made known all of his discoveries, mathematics would have been advanced by 50 years.
A criticism of Gauss is that he didn't 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 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. This is fully, however briefly, explained by Gauss himself in his "Disquisitiones Arithmeticae", where he states that all analysis (for example the paths one travelled to reach the solution of a problem) must be suppressed for sake of brevity.
Gauss was deeply religious and conservative. He supported monarchy and opposed
Napoleon, whom he saw as an outgrowth of
revolution.
Commemorations
The
CGS unit for
magnetic induction was named
gauss in his honour.
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
Hanover. 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.
In 2007, his
bust was introduced to the
Walhalla temple.
Places, vessels and events named in honour of Gauss:
- Gauss crater on the Moon
- 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 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
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