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117-593: Leonhard may refer to: Leonhard Euler (1707–1783), Swiss mathematician and physicist Leonhard Hutter (1563–1616), German theologian Karl Leonhard (1904–1988), German psychiatrist Jim Leonhard (1982– ), American football safety Leonhard Rauwolf (1535–1596), German physician and botanist Leonhard Stejneger (1851–1943), American herpetologist Wolfgang Leonhard (1921–2014), German author & historian See also [ edit ] Leonard#Variations Leonard (disambiguation) Topics referred to by
234-671: A Doctor of Philosophy in 1799, not in Göttingen, as is sometimes stated, but at the Duke of Brunswick's special request from the University of Helmstedt, the only state university of the duchy. Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degree in absentia without further oral examination. The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from
351-522: A convex polyhedron , and hence of a planar graph . The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. The study and generalization of this formula, specifically by Cauchy and L'Huilier , is at the origin of topology . Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of
468-492: A heliometer from Fraunhofer . The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade. Gauss made no secret of his aversion to giving academic lectures. But from the start of his academic career at Göttingen, he continuously gave lectures until 1854. He often complained about
585-580: A German Princess . This work contained Euler's exposition on various subjects pertaining to physics and mathematics and offered valuable insights into Euler's personality and religious beliefs. It was translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. The popularity of the Letters testifies to Euler's ability to communicate scientific matters effectively to
702-488: A basis for Gauss's research on their orbits, which he later published in his astronomical magnum opus Theoria motus corporum coelestium (1809). In November 1807, Gauss followed a call to the University of Göttingen , then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte , as full professor and director of the astronomical observatory , and kept the chair until his death in 1855. He
819-491: A considerable literary estate, too. Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician. On certain occasions, Gauss claimed that the ideas of another scholar had already been in his possession previously. Thus his concept of priority as "the first to discover, not
936-492: A converted fortification tower, with usable, but partly out-of-date instruments. The construction of a new observatory had been approved by Prince-elector George III in principle since 1802, and the Westphalian government continued the planning, but Gauss could not move to his new place of work until September 1816. He got new up-to-date instruments, including two meridian circles from Repsold and Reichenbach , and
1053-672: A critique of d'Alembert's work. 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. In the preface to the Disquisitiones , Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had discovered by himself. The Disquisitiones Arithmeticae , written since 1798 and published in 1801, consolidated number theory as
1170-431: A curious feature of his working style that he carried out calculations with a high degree of precision much more than required, and prepared tables with more decimal places than ever requested for practical purposes. Very likely, this method gave him a lot of material which he used in finding theorems in number theory. Gauss refused to publish work that he did not consider complete and above criticism. This perfectionism
1287-474: A decade. Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau. Her sister Wilhelmina married the orientalist Heinrich Ewald . Gauss's mother Dorothea lived in his house from 1817 until she died in 1839. The eldest son Joseph, while still a schoolboy, helped his father as an assistant during the survey campaign in
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#17327909772791404-483: A discipline and covered both elementary and algebraic number theory . Therein he introduces the triple bar symbol ( ≡ ) for congruence and uses it for a clean presentation of modular arithmetic . It deals with the unique factorization theorem and primitive roots modulo n . In the main sections, Gauss presents the first two proofs of the law of quadratic reciprocity and develops the theories of binary and ternary quadratic forms . The Disquisitiones include
1521-586: A geometrical problem that had occupied mathematicians since the Ancient Greeks , when he determined in 1796 which regular polygons can be constructed by compass and straightedge . This discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss's mathematical diary, a collection of short remarks about his results from the years 1796 until 1814, shows that many ideas for his mathematical magnum opus Disquisitiones Arithmeticae (1801) date from this time. Gauss graduated as
1638-682: A habit in his later years, for example, the number of paths from his home to certain places in Göttingen, or the number of living days of persons; he congratulated Humboldt in December 1851 for having reached the same age as Isaac Newton at his death, calculated in days. Similar to his excellent knowledge of Latin he was also acquainted with modern languages. At the age of 62, he began to teach himself Russian , very likely to understand scientific writings from Russia, among them those of Lobachevsky on non-Euclidean geometry. Gauss read both classical and modern literature, and English and French works in
1755-555: A heart attack in Göttingen; and was interred in the Albani Cemetery there. Heinrich Ewald , Gauss's son-in-law, and Wolfgang Sartorius von Waltershausen , Gauss's close friend and biographer, gave eulogies at his funeral. Gauss was a successful investor and accumulated considerable wealth with stocks and securities, finally a value of more than 150 thousand Thaler; after his death, about 18 thousand Thaler were found hidden in his rooms. The day after Gauss's death his brain
1872-449: A lay audience, a rare ability for a dedicated research scientist. Despite Euler's immense contribution to the academy's prestige and having been put forward as a candidate for its presidency by Jean le Rond d'Alembert , Frederick II named himself as its president. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler
1989-469: A lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with Anders Johan Lexell when he collapsed and died from a brain hemorrhage . Jacob von Staehlin [ de ] wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss , one of Euler's disciples, wrote a more detailed eulogy, which he delivered at
2106-567: A memorial meeting. In his eulogy for the French Academy , French mathematician and philosopher Marquis de Condorcet , wrote: il cessa de calculer et de vivre — ... he ceased to calculate and to live. Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island . In 1837, the Russian Academy of Sciences installed a new monument, replacing his overgrown grave plaque. To commemorate
2223-469: A new field of study, analytic number theory . In breaking ground for this new field, Euler created the theory of hypergeometric series , q-series , hyperbolic trigonometric functions , and the analytic theory of continued fractions . For example, he proved the infinitude of primes using the divergence of the harmonic series , and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to
2340-588: A pension for his wife, and the promise of high-ranking appointments for his sons. At the university he was assisted by his student Anders Johan Lexell . While living in St. Petersburg, a fire in 1771 destroyed his home. On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell , a painter from the Academy Gymnasium in Saint Petersburg. The young couple bought a house by
2457-462: A result otherwise known as the Euclid–Euler theorem . Euler also conjectured the law of quadratic reciprocity . The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of Carl Friedrich Gauss , particularly Disquisitiones Arithmeticae . By 1772 Euler had proved that 2 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained
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#17327909772792574-639: A scandal in public, Eugen suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken to start, after which his father refused further financial support. The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties getting an appropriate education, and eventually emigrated as well. Only Gauss's youngest daughter Therese accompanied him in his last years of life. Collecting numerical data on very different things, useful or useless, became
2691-415: A strong calculus as the sole tasks of astronomy. At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program; these included the mathematician Thibaut with his lectures, the physicist Mayer , known for his textbooks, his successor Weber since 1831, and in the observatory Harding , who took the main part of lectures in practical astronomy. When
2808-573: A technical perspective. Euler's calculations look likely to be correct, even if Euler's interactions with Frederick and those constructing his fountain may have been dysfunctional. Throughout his stay in Berlin, Euler maintained a strong connection to the academy in St. Petersburg and also published 109 papers in Russia. He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin. In 1760, with
2925-445: A university chair in Göttingen, "because he was always involved in some polemic." Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancient threnody , the most personal surviving document of Gauss. The situation worsened when tuberculosis ultimately destroyed
3042-478: A water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci . My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry! However, the disappointment was almost surely unwarranted from
3159-501: A way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis . He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations . Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced
3276-490: Is associated with a large number of topics . Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonhard Euler was the author of a quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century. Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced
3393-455: Is different from Wikidata All article disambiguation pages All disambiguation pages Leonhard Euler Leonhard Euler ( / ˈ ɔɪ l ər / OY -lər ; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleɔnhard ˈɔʏlər] ; 15 April 1707 – 18 September 1783) was a Swiss mathematician , physicist , astronomer , geographer , logician , and engineer who founded
3510-481: Is regarded as one of the greatest, most prolific mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain
3627-430: Is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit . This solution is considered to be the first theorem of graph theory . Euler also discovered the formula V − E + F = 2 {\displaystyle V-E+F=2} relating the number of vertices, edges, and faces of
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3744-640: The Introductio in analysin infinitorum was published and in 1755 a text on differential calculus called the Institutiones calculi differentialis was published. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences and of the French Academy of Sciences . Notable students of Euler in Berlin included Stepan Rumovsky , later considered as the first Russian astronomer. In 1748 he declined an offer from
3861-1040: The Basel problem , finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by Pietro Mengoli in 1644, and by the 1730s was a famous open problem, popularized by Jacob Bernoulli and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that: ∑ n = 1 ∞ 1 n 2 = lim n → ∞ ( 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1 n 2 ) = π 2 6 . {\displaystyle \sum _{n=1}^{\infty }{1 \over n^{2}}=\lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}.} Euler introduced
3978-459: The Bernoulli numbers , Fourier series , Euler numbers , the constants e and π , continued fractions, and integrals. He integrated Leibniz 's differential calculus with Newton's Method of Fluxions , and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as
4095-538: The Euler approximations . The most notable of these approximations are Euler's method and the Euler–Maclaurin formula . Carl Friedrich Gauss Johann Carl Friedrich Gauss (German: Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs] ; Latin : Carolus Fridericus Gauss ; 30 April 1777 – 23 February 1855) was a German mathematician , astronomer , geodesist , and physicist who contributed to many fields in mathematics and science. He
4212-502: The Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares , he proves the triangular case of the Fermat polygonal number theorem for n = 3. From several analytic results on class numbers that Gauss gives without proof towards
4329-482: The Neva River . Of their thirteen children, only five survived childhood, three sons and two daughters. Their first son was Johann Albrecht Euler , whose godfather was Christian Goldbach . Three years after his wife's death in 1773, Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death in 1783. His brother Johann Heinrich settled in St. Petersburg in 1735 and
4446-474: The Russian Academy of Sciences in St. Peterburg and Landshut University . Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans: the Duke was killed in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped. When Gauss
4563-569: The Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, with Empress Elizabeth of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time. Euler decided to leave Berlin in 1766 and return to Russia. During his Berlin years (1741–1766), Euler
4680-456: The cartography he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains the subject of speculation. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as " Cyclops ". Euler remarked on his loss of vision, stating "Now I will have fewer distractions." In 1766 a cataract in his left eye
4797-527: The largest known prime until 1867. Euler also contributed major developments to the theory of partitions of an integer . In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg . The city of Königsberg , Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem
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4914-550: The popularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay Erdmagnetismus und Magnetometer of 1836. Gauss published his papers and books exclusively in Latin or German . He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians. In his inaugural lecture at Göttingen University from 1808, Gauss claimed reliable observations and results attained only by
5031-554: The propagation of sound with the title De Sono with which he unsuccessfully attempted to obtain a position at the University of Basel. In 1727, he entered the Paris Academy prize competition (offered annually and later biennially by the academy beginning in 1720) for the first time. The problem posed that year was to find the best way to place the masts on a ship. Pierre Bouguer , who became known as "the father of naval architecture", won and Euler took second place. Over
5148-477: The ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones . The development of infinitesimal calculus was at the forefront of 18th-century mathematical research, and the Bernoullis —family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became
5265-692: The 250th anniversary of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery . Euler worked in almost all areas of mathematics, including geometry , infinitesimal calculus , trigonometry , algebra , and number theory , as well as continuum physics , lunar theory , and other areas of physics . He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler's name
5382-500: The Duke granted him the resources for studies of mathematics, sciences, and classical languages at the University of Göttingen until 1798. His professor in mathematics was Abraham Gotthelf Kästner , whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his epigrams . Astronomy was taught by Karl Felix Seyffer , with whom Gauss stayed in correspondence after graduation; Olbers and Gauss mocked him in their correspondence. On
5499-540: The Euler–Mascheroni constant, and studied its relationship with the harmonic series , the gamma function , and values of the Riemann zeta function . Euler introduced the use of the exponential function and logarithms in analytic proofs . He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers , thus greatly expanding
5616-494: The French language. Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration. Whereas previous mathematicians like Leonhard Euler let the readers take part in their reasoning for new ideas, including certain erroneous deviations from the correct path, Gauss however introduced a new style of direct and complete explanation that did not attempt to show
5733-497: The Greek letter Σ {\displaystyle \Sigma } (capital sigma ) to express summations , the Greek letter Δ {\displaystyle \Delta } (capital delta ) for finite differences , and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant e {\displaystyle e} ,
5850-454: The Greek letter π {\displaystyle \pi } (lowercase pi ) to denote the ratio of a circle's circumference to its diameter , as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\displaystyle {\sqrt {-1}}} ,
5967-681: The Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics. In 1720, at thirteen years of age, Euler enrolled at the University of Basel . Attending university at such a young age was not unusual at the time. The course on elementary mathematics was given by Johann Bernoulli , the younger brother of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography: It
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#17327909772796084-939: The Russian Navy, refusing a promotion to lieutenant . Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell . Frederick II had made an attempt to recruit the services of Euler for his newly established Berlin Academy in 1740, but Euler initially preferred to stay in St Petersburg. But after Empress Anna died and Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave to Berlin, arguing he
6201-689: The University of Basel to succeed the recently deceased Johann Bernoulli. In 1753 he bought a house in Charlottenburg , in which he lived with his family and widowed mother. Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt , the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to
6318-549: The academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and universities. Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia assumed power. Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. He also left
6435-475: The act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The posthumous papers, his scientific diary , and short glosses in his own textbooks show that he worked to a great extent in an empirical way. He was a lifelong busy and enthusiastic calculator, who made his calculations with extraordinary rapidity, mostly without precise controlling, but checked
6552-439: The base of the natural logarithm , now known as Euler's number . Euler is also credited with being the first to develop graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg , which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, providing a solution to several unsolved problems in number theory and analysis, including
6669-596: The best school for the different fields of mathematics, and nothing else can replace it." His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos . He spent most of his adult life in Saint Petersburg , Russia, and in Berlin , then the capital of Prussia . Euler is credited for popularizing
6786-637: The best-paid professors of the university. When Gauss was asked for help by his colleague and friend Friedrich Wilhelm Bessel in 1810, who was in trouble at Königsberg University because of his lack of an academic title, Gauss provided a doctorate honoris causa for Bessel from the Philosophy Faculty of Göttingen in March 1811. Gauss gave another recommendation for an honorary degree for Sophie Germain but only shortly before her death, so she never received it. He also gave successful support to
6903-511: The birth of Leonhard, the Euler family moved from Basel to the town of Riehen , Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood. From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some years earlier at the University of Basel . Around the age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in
7020-517: The birth of Louis, who himself died a few months later. Gauss chose the first names of his children in honour of Giuseppe Piazzi , Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids. On 4 August 1810, Gauss married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he had three more children: Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879), and Therese (1816–1864). Minna Gauss died on 12 September 1831 after being seriously ill for more than
7137-701: The burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described some students as talented. Most of his lectures dealt with astronomy, geodesy, and applied mathematics , and only three lectures on subjects of pure mathematics. Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor , Dedekind , Dirksen , Encke , Gould , Heine , Klinkerfues , Kupffer , Listing , Möbius , Nicolai , Riemann , Ritter , Schering , Scherk , Schumacher , von Staudt , Stern , Ursin ; as geoscientists Sartorius von Waltershausen , and Wappäus . Gauss did not write any textbook and disliked
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#17327909772797254-423: The concept of a function and was the first to write f ( x ) to denote the function f applied to the argument x . He also introduced the modern notation for the trigonometric functions , the letter e for the base of the natural logarithm (now also known as Euler's number ), the Greek letter Σ for summations and the letter i to denote the imaginary unit . The use of the Greek letter π to denote
7371-498: The constant γ = lim n → ∞ ( 1 + 1 2 + 1 3 + 1 4 + ⋯ + 1 n − ln ( n ) ) ≈ 0.5772 , {\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{n}}-\ln(n)\right)\approx 0.5772,} now known as Euler's constant or
7488-496: The contemporary school of Naturphilosophie . Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto " mundus vult decipi ". He disliked Napoleon and his system, and all kinds of violence and revolution caused horror to him. Thus he condemned the methods of the Revolutions of 1848 , though he agreed with some of their aims, such as
7605-537: The development of the prime number theorem . Euler's interest in number theory can be traced to the influence of Christian Goldbach , his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre de Fermat . Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form 2 2 n + 1 {\textstyle 2^{2^{n}}+1} ( Fermat numbers ) are prime. Euler linked
7722-412: The end of the fifth section, it appears that Gauss already knew the class number formula in 1801. In the last section, Gauss gives proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical problem to an algebraic one. He shows that a regular polygon is constructible if the number of its sides is either a power of 2 or
7839-451: The famous Basel problem . Euler has also been credited for discovering that the sum of the numbers of vertices and faces minus the number of edges of a polyhedron equals 2, a number now commonly known as the Euler characteristic . In the field of physics, Euler reformulated Newton 's laws of physics into new laws in his two-volume work Mechanica to better explain the motion of rigid bodies . He also made substantial contributions to
7956-621: The first investigations, due to mislabelling, with that of the physician Conrad Heinrich Fuchs , who died in Göttingen a few months after Gauss. A further investigation showed no remarkable anomalies in the brains of both persons. Thus, all investigations on Gauss's brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs. Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick. They had two sons and one daughter: Joseph (1806–1873), Wilhelmina (1808–1840), and Louis (1809–1810). Johanna died on 11 October 1809, one month after
8073-440: The first to publish" differed from that of his scientific contemporaries. In contrast to his perfectionism in presenting mathematical ideas, he was criticized for a negligent way of quoting. He justified himself with a very special view of correct quoting: if he gave references, then only in a quite complete way, with respect to the previous authors of importance, which no one should ignore; but quoting in this way needed knowledge of
8190-525: The health of his second wife Minna over 13 years; both his daughters later suffered from the same disease. Gauss himself gave only slight hints of his distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings". By reason of his wife's illness, both younger sons were educated for some years in Celle , far from Göttingen. The military career of his elder son Joseph ended after more than two decades with
8307-460: The history of science and more time than he wished to spend. Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty, but also of "iron character" with an unshakeable strength of mind. Apart from his closer circle, others regarded him as reserved and unapproachable "like an Olympian sitting enthroned on
8424-467: The idea of a unified Germany. As far as the political system is concerned, he had a low estimation of the constitutional system; he criticized parliamentarians of his time for a lack of knowledge and logical errors. Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes" and "I succeeded – not on account of my hard efforts, but by the grace of the Lord." Gauss
8541-414: The integer n that are coprime to n . Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem . He contributed significantly to the theory of perfect numbers , which had fascinated mathematicians since Euclid . He proved that the relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one,
8658-1007: The major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra ), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series , the expression of functions as sums of infinitely many terms, such as e x = ∑ n = 0 ∞ x n n ! = lim n → ∞ ( 1 0 ! + x 1 ! + x 2 2 ! + ⋯ + x n n ! ) . {\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\lim _{n\to \infty }\left({\frac {1}{0!}}+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{n}}{n!}}\right).} Euler's use of power series enabled him to solve
8775-545: The mathematician Gotthold Eisenstein in Berlin. Gauss was loyal to the House of Hanover . After King William IV died in 1837, the new Hanoverian King Ernest Augustus annulled the 1833 constitution. Seven professors, later known as the " Göttingen Seven ", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss's son-in-law Heinrich Ewald. All of them were dismissed, and three of them were expelled, but Ewald and Weber could stay in Göttingen. Gauss
8892-457: The mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726, Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg in May 1727. He was promoted from his junior post in
9009-524: The medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as a medic in the Russian Navy . The academy at Saint Petersburg, established by Peter the Great , was intended to improve education in Russia and to close
9126-455: The nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges . In doing so, he discovered the connection between the Riemann zeta function and prime numbers; this is known as the Euler product formula for the Riemann zeta function . Euler invented the totient function φ( n ), the number of positive integers less than or equal to
9243-590: The observatory was completed, Gauss took his living accommodation in the western wing of the new observatory and Harding in the eastern one. They had once been on friendly terms, but over time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer. Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations. Brendel subdivides Gauss's astronomic activity chronologically into seven periods, of which
9360-520: The observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income. He was even involved in the design of the water fountains at Sanssouci , the King's summer palace. The political situation in Russia stabilized after Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary,
9477-411: The original languages. His favorite English author was Walter Scott , his favorite German Jean Paul . Gauss liked singing and went to concerts. He was a busy newspaper reader; in his last years, he used to visit an academic press salon of the university every noon. Gauss did not care much for philosophy, and mocked the "splitting hairs of the so-called metaphysicians", by which he meant proponents of
9594-416: The other hand, he thought highly of Georg Christoph Lichtenberg , his teacher of physics, and of Christian Gottlob Heyne , whose lectures in classics Gauss attended with pleasure. Fellow students of this time were Johann Friedrich Benzenberg , Farkas Bolyai , and Heinrich Wilhelm Brandes . He was likely a self-taught student in mathematics since he independently rediscovered several theorems. He solved
9711-475: The problem by accepting offers from Berlin in 1810 and 1825 to become a full member of the Prussian Academy without burdening lecturing duties, as well as from Leipzig University in 1810 and from Vienna University in 1842, perhaps because of the family's difficult situation. Gauss's salary was raised from 1000 Reichsthaler in 1810 to 2400 Reichsthaler in 1824, and in his later years he was one of
9828-425: The product of a power of 2 and any number of distinct Fermat primes . In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in finite fields , which amounts to counting integral points on an elliptic curve . An unfinished eighth chapter was found among left papers only after his death, consisting of work done during 1797–1799. One of Gauss's first results
9945-592: The railroad system in the US for some months. Eugen left Göttingen in September 1830 and emigrated to the United States, where he joined the army for five years. He then worked for the American Fur Company in the Midwest. Later, he moved to Missouri and became a successful businessman. Wilhelm married a niece of the astronomer Bessel ; he then moved to Missouri, started as a farmer and became wealthy in
10062-437: The rank of a poorly paid first lieutenant , although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married. The second son Eugen shared a good measure of his father's talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused
10179-435: The reader the author's train of thought. Gauss was the first to restore that rigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period in new developments. But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows: It is not knowledge, but the act of learning, not possession but
10296-466: The results by masterly estimation. Nevertheless, his calculations were not always free from mistakes. He coped with the enormous workload by using skillful tools. Gauss used a lot of mathematical tables , examined their exactness, and constructed new tables on various matters for personal use. He developed new tools for effective calculation, for example the Gaussian elimination . It has been taken as
10413-413: The same term [REDACTED] This disambiguation page lists articles associated with the title Leonhard . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Leonhard&oldid=1253309638 " Category : Disambiguation pages Hidden categories: Short description
10530-411: The scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy's benefactress, Catherine I , who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg. The Russian conservative nobility then gained power upon the ascension of the twelve-year-old Peter II . The nobility, suspicious of
10647-504: The scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions . For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies e i φ = cos φ + i sin φ {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi } which
10764-498: The second and third complete proofs of the fundamental theorem of algebra , made contributions to number theory , and developed the theories of binary and ternary quadratic forms. Gauss was instrumental in the identification of Ceres as a dwarf planet. His work on the motion of planetoids disturbed by large planets led to the introduction of the Gaussian gravitational constant and the method of least squares , which he had discovered before Adrien-Marie Legendre published it. Gauss
10881-558: The shoe business in St. Louis in later years. Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children. In the first two decades of the 19th century, Gauss was the only important mathematician in Germany, comparable to the leading French ones; his Disquisitiones Arithmeticae was the first mathematical book from Germany to be translated into
10998-452: The studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory , complex analysis , and infinitesimal calculus . He introduced much of modern mathematical terminology and notation , including the notion of a mathematical function . He is also known for his work in mechanics , fluid dynamics , optics , astronomy , and music theory . Euler
11115-479: The study of elastic deformations of solid objects. Leonhard Euler was born on 15 April 1707, in Basel to Paul III Euler, a pastor of the Reformed Church , and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics. He was the oldest of four children, having two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich. Soon after
11232-577: The summer of 1821. After a short time at university, in 1824 Joseph joined the Hanoverian army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network to the western parts of the kingdom. With his geodetical qualifications, he left the service and engaged in the construction of the railway network as director of the Royal Hanoverian State Railways . In 1836 he studied
11349-421: The summit of science". His close contemporaries agreed that Gauss was a man of difficult character. He often refused to accept compliments. His visitors were occasionally irritated by his grumpy behaviour, but a short time later his mood could change, and he would become a charming, open-minded host. Gauss abominated polemic natures; together with his colleague Hausmann he opposed to a call for Justus Liebig on
11466-421: The years since 1820 are taken as a "period of lower astronomical activity". The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834. Nevertheless, Gauss twice refused the opportunity to solve
11583-534: The years, Euler entered this competition 15 times, winning 12 of them. Johann Bernoulli's two sons, Daniel and Nicolaus , entered into service at the Imperial Russian Academy of Sciences in Saint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. When Daniel assumed his brother's position in
11700-792: Was a member of the Lutheran church , like most of the population in northern Germany. It seems that he did not believe all dogmas or understand the Holy Bible quite literally. Sartorius mentioned Gauss's religious tolerance , and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions. In his doctoral thesis from 1799, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root . Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains
11817-491: Was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways, the polar opposite of Voltaire , who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit. Frederick also expressed disappointment with Euler's practical engineering abilities, stating: I wanted to have
11934-541: Was at the peak of his productivity. He wrote 380 works, 275 of which were published. This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to the St. Petersburg Academy , which had retained him as a member and paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in two parts in 1748. In addition to his own research, Euler supervised the library,
12051-570: Was born on 30 April 1777 in Brunswick in the Duchy of Brunswick-Wolfenbüttel (now in the German state of Lower Saxony ). His family was of relatively low social status. His father Gebhard Dietrich Gauss (1744–1808) worked variously as a butcher, bricklayer, gardener, and treasurer of a death-benefit fund. Gauss characterized his father as honourable and respected, but rough and dominating at home. He
12168-473: Was calculating asteroid orbits in the first years of the century, he established contact with the astronomical community of Bremen and Lilienthal , especially Wilhelm Olbers , Karl Ludwig Harding , and Friedrich Wilhelm Bessel , as part of the informal group of astronomers known as the Celestial police . One of their aims was the discovery of further planets. They assembled data on asteroids and comets as
12285-427: Was called "the most remarkable formula in mathematics" by Richard Feynman . A special case of the above formula is known as Euler's identity , e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations . He found
12402-701: Was deeply affected by this quarrel but saw no possibility to help them. Gauss took part in academic administration: three times he was elected as dean of the Faculty of Philosophy. Being entrusted with the widow's pension fund of the university, he dealt with actuarial science and wrote a report on the strategy for stabilizing the benefits. He was appointed director of the Royal Academy of Sciences in Göttingen for nine years. Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. On 23 February 1855, he died of
12519-477: Was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. He is widely considered one of the greatest mathematicians ever. While studying at the University of Göttingen , he propounded several mathematical theorems . Gauss completed his masterpieces Disquisitiones Arithmeticae and Theoria motus corporum coelestium as a private scholar. He gave
12636-451: Was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in the left eye as well. However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased; and, in 1775, he produced, on average, one mathematical paper every week. In St. Petersburg on 18 September 1783, after
12753-400: Was during this time that Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor. In 1723, Euler received a Master of Philosophy with a dissertation that compared the philosophies of René Descartes and Isaac Newton . Afterwards, he enrolled in the theological faculty of the University of Basel. In 1726, Euler completed a dissertation on
12870-511: Was employed as a painter at the academy. Early in his life, Euler memorized the entirety of the Aeneid by Virgil , and by old age, could recite the entirety of the poem, along with stating the first and last sentence on each page of the edition from which he had learnt it. Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye. Euler blamed
12987-578: Was experienced in writing and calculating, whereas his second wife Dorothea, Carl Friedrich's mother, was nearly illiterate. He had one elder brother from his father's first marriage. Gauss was a child prodigy in mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick who sent him to the local Collegium Carolinum , which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter
13104-487: Was in charge of the extensive geodetic survey of the Kingdom of Hanover together with an arc measurement project from 1820 to 1844; he was one of the founders of geophysics and formulated the fundamental principles of magnetism . Fruits of his practical work were the inventions of the heliotrope in 1821, a magnetometer in 1833 and – alongside Wilhelm Eduard Weber – the first electromagnetic telegraph in 1833. Gauss
13221-517: Was in keeping with the motto of his personal seal Pauca sed Matura ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself, claiming that the initial discovery of ideas was easy, but preparing a presentable elaboration was a demanding matter for him, for either lack of time or "serenity of mind". Nevertheless, he published many short communications of urgent content in various journals, but left
13338-582: Was in need of a milder climate for his eyesight. The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy , which he had been offered by Frederick the Great of Prussia . He lived for 25 years in Berlin , where he wrote several hundred articles. In 1748 his text on functions called
13455-512: Was removed, preserved, and studied by Rudolf Wagner , who found its mass to be slightly above average, at 1,492 grams (3.29 lb). Wagner's son Hermann , a geographer, estimated the cerebral area to be 219,588 square millimetres (340.362 sq in) in his doctoral thesis. In 2013, a neurobiologist at the Max Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after
13572-488: Was soon confronted with the demand for two thousand francs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and Laplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from Frankfurt , later discovered to be Prince-primate Dalberg , paid the sum. Gauss took on the directorate of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on
13689-543: Was the first to discover and study non-Euclidean geometry , coining the term as well. He further developed a fast Fourier transform some 160 years before John Tukey and James Cooley . Gauss refused to publish incomplete work and left several works to be edited posthumously . He believed that the act of learning, not possession of knowledge, provided the greatest enjoyment. Gauss confessed to disliking teaching, but some of his students became influential mathematicians, such as Richard Dedekind and Bernhard Riemann . Gauss
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