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119-515: Jacob Bernoulli was born in Basel in the Old Swiss Confederacy . Following his father's wish, he studied theology and entered the ministry. But contrary to the desires of his parents, he also studied mathematics and astronomy . He traveled throughout Europe from 1676 to 1682, learning about the latest discoveries in mathematics and the sciences under leading figures of the time. This included

238-518: 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

357-415: A logarithmic spiral and the motto Eadem mutata resurgo ('Although changed, I rise again the same') engraved on his tombstone. He wrote that the self-similar spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self". Bernoulli died in 1705, but an Archimedean spiral

476-568: 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

595-401: A bishops' crozier as the heraldic charge in the coat of arms of Basel first appears in the form of a gilded wooden staff in the 12th century. It is of unknown origin or significance (beyond its obvious status of bishop's crozier), but it is assumed to have represented a relic, possibly attributed to Saint Germanus of Granfelden . This staff (known as Baselstab ) became a symbol representing

714-427: A century earlier. For many centuries to come Basel possessed the only permanent bridge over the river "between Lake Constance and the sea". The first city guild were the furriers , established in 1226. A total of about fifteen guilds were established in the course of the 13th century, reflecting the increasing economic prosperity of the city. The Crusade of 1267 set out from Basel. Political conflicts between

833-456: A commercial hub and an important cultural centre since the Renaissance , and has emerged as a centre for the chemical and pharmaceutical industries in the 20th century. In 1897, Basel was chosen by Theodor Herzl as the location for the first World Zionist Congress , and altogether the congress was held there ten times over a time span of 50 years, more than in any other location. The city

952-507: A fertile research career. His travels allowed him to establish correspondence with many leading mathematicians and scientists of his era, which he maintained throughout his life. During this time, he studied the new discoveries in mathematics, including Christiaan Huygens 's De ratiociniis in aleae ludo , Descartes ' La Géométrie and Frans van Schooten 's supplements of it. He also studied Isaac Barrow and John Wallis , leading to his interest in infinitesimal geometry. Apart from these, it

1071-491: A finite limit less than 2. Euler was the first to find the limit of this series in 1737. Bernoulli also studied the exponential series which came out of examining compound interest. In May 1690, in a paper published in Acta Eruditorum , Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent,

1190-524: A handful of wealthy families collectively referred to as the "Daig" played a pivotal role in city affairs as they gradually established themselves as a de facto city aristocracy . The first edition of Christianae religionis institutio ( Institutes of the Christian Religion – John Calvin 's great exposition of Calvinist doctrine) was published at Basel in March 1536. In 1544, Johann von Brugge,

1309-449: 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

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1428-600: A limit (the force of interest ) for more and smaller compounding intervals. Compounding weekly yields $ 2.692597..., while compounding daily yields $ 2.714567..., just two cents more. Using n as the number of compounding intervals, with interest of 100% / n in each interval, the limit for large n is the number that Euler later named e ; with continuous compounding, the account value will reach $ 2.7182818.... More generally, an account that starts at $ 1, and yields (1+ R ) dollars at compound interest , will yield e dollars with continuous compounding. Bernoulli wanted

1547-462: 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

1666-549: 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

1785-469: 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

1904-496: A pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines. By 1689, he had published important work on infinite series and published his law of large numbers in probability theory. Jacob Bernoulli published five treatises on infinite series between 1682 and 1704. The first two of these contained many results, such as

2023-569: 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

2142-508: A pro-Habsburg faction, known as Sterner , and an anti-Habsburg faction, the Psitticher . The Black Death reached Basel in 1348. The Jews were blamed , and an estimated 50 to 70 Jews were executed by burning on 16 January 1349 in what has become known as the Basel massacre . The Basel earthquake of 1356 destroyed much of the city along with a number of castles in the vicinity. A riot on 26 February 1376, known as Böse Fasnacht , led to

2261-462: 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

2380-556: A rich Dutch Protestant refugee, was given citizenship and lived respectably until his death in 1556, then buried with honors. His body was exhumed and burnt at the stake in 1559 after it was discovered that he was the Anabaptist David Joris . In 1543, De humani corporis fabrica , the first book on human anatomy, was published and printed in Basel by Andreas Vesalius (1514–1564). There are indications Joachim Meyer , author of

2499-565: 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

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2618-473: 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

2737-501: 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

2856-522: Is May during which time Basel receives an average of 98 mm (3.9 in) of rain. The month with the most days of precipitation is also May, with an average of 11.7 days. The driest month of the year is February with an average of 45 mm (1.8 in) of precipitation over 8.4 days. The city of Basel functions as the capital of the Swiss half-canton of Basel-Stadt . The canton Basel-Stadt consists of three municipalities: Riehen , Bettingen , and

2975-521: Is also home to the world headquarters of the Bank for International Settlements . The name of the city is internationally known through institutions like the Basel Accords , Art Basel and FC Basel . Basel is Switzerland's main centre for the pharmaceutical industry, hosting both Novartis and Roche . In 2019 Basel was ranked the tenth most liveable city in the world by Mercer . The name of Basel

3094-490: 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

3213-584: Is first recorded as Basilia in the 3rd century (237/8), at the time referring to the Roman castle . This name is mostly interpreted as deriving from the personal name Basilius , from a toponym villa Basilia (" estate of Basilius") or similar. Another suggestion derives it from a name Basilia attested in northern France as a development of basilica , the term for a public or church building (as in Bazeilles ), but all of these names reference early church buildings of

3332-446: Is forested. Of the rest of the land, 20.67 km (7.98 sq mi) or 86.4% is settled (buildings or roads), 1.45 km (0.56 sq mi) or 6.1% is either rivers or lakes. Of the built up area, industrial buildings made up 10.2% of the total area while housing and buildings made up 40.7% and transportation infrastructure made up 24.0%. Power and water infrastructure as well as other special developed areas made up 2.7% of

3451-510: Is in fact e ): One example is an account that starts with $ 1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $ 2.00; but if the interest is computed and added twice in the year, the $ 1 is multiplied by 1.5 twice, yielding $ 1.00×1.5 = $ 2.25. Compounding quarterly yields $ 1.00×1.25 = $ 2.4414..., and compounding monthly yields $ 1.00×(1.0833...) = $ 2.613035.... Bernoulli noticed that this sequence approaches

3570-708: Is located in Northwestern Switzerland and is commonly considered to be the capital of that region. It is close to the point where the Swiss, French and German borders meet, and Basel also has suburbs in France and Germany. As of 2016 , the Swiss Basel agglomeration was the third-largest in Switzerland, with a population of 541,000 in 74 municipalities in Switzerland (municipal count as of 2018). The metropolitan area , called

3689-459: Is recorded for 1185, and the first mayor , Heinrich Steinlin of Murbach, for 1253. The first bridge across the Rhine was built in 1225 under bishop Heinrich von Thun (at the location of the modern Middle Bridge ), and from this time the settlement of Kleinbasel gradually formed around the bridgehead on the far river bank. The bridge was largely funded by Basel's Jewish community who had settled there

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3808-481: 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

3927-405: Is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. It had been studied by Huygens in 1687 and Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call separation of variables . Jacob Bernoulli's paper of 1690 is important for the history of calculus, since

4046-430: 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

4165-618: 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

4284-461: The BBC . Currently, the spelling Basel is most often used, to match the official German spelling. In French Basle was still in use in the 18th century, but was gradually replaced by the modern French spelling Bâle . In Icelandic, the city is recorded as Buslaraborg in the 12th-century itinerary Leiðarvísir og borgarskipan . There are traces of a settlement at the nearby Rhine knee from

4403-989: 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

4522-455: The Baselstab in red represented the bishop, and the same charge in black represented the city. The blazon of the municipal coat of arms is In Silber ein schwarzer Baselstab (Argent, a staff of Basel sable). In 1400, Basel was able to purchase the towns of Liestal , Homburg and Waldenburg with its surrounding territory. In 1412 (or earlier), the well-known Gasthof zum Goldenen Sternen

4641-454: 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

4760-670: The Fondation Beyeler (located in Riehen ), the Museum Tinguely and the Museum of Contemporary Art , which is the first public museum of contemporary art in Europe. Forty museums are spread throughout the city-canton, making Basel one of the largest cultural centres in relation to its size and population in Europe. The University of Basel , Switzerland's oldest university (founded in 1460), and

4879-701: The French Revolutionary Wars . In more recent times, the World Zionist Organization held its first congress in Basel from 29 August through 31 August 1897. Because of the Balkan Wars , the (Socialist) Second International held an extraordinary congress at Basel in 1912. In 1989, the Basel Convention was opened for signature with the aim of preventing the export of hazardous waste from wealthy to developing nations for disposal. Basel

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4998-476: 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

5117-469: The Roman invasion of Gaul . In Roman Gaul , Augusta Raurica was established some 20 km (12 mi) from Basel as the regional administrative centre, while a castrum (fortified camp) was built on the site of the Celtic oppidum . In AD 83, the area was incorporated into the Roman province of Germania Superior . The Roman Senator Munatius Plancus is known as the traditional founder of Basel since

5236-556: 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

5355-590: The Swabian League of Cities in 1385, and many knights of the pro-Habsburg faction, along with duke Leopold himself, were killed in the Battle of Sempach the following year. A formal treaty with Habsburg was made in 1393. Basel had gained its de facto independence from both the bishop and from the Habsburgs and was free to pursue its own policy of territorial expansion, beginning around 1400. The unique representation of

5474-688: The Treaty of Meerssen of 870. Basel was destroyed by the Magyars in 917. The rebuilt town became part of Upper Burgundy , and as such was incorporated into the Holy Roman Empire in 1032. From the donation by Rudolph III of Burgundy of the Moutier-Grandval Abbey and all its possessions to Bishop Adalbero II of Metz in 999 until the Reformation , Basel was ruled by Prince-Bishops . In 1019,

5593-468: The Trinational Eurodistrict of Basel (TEB), consists of 62 suburban communes including municipalities in neighboring countries, and counted 829,000 inhabitants in 2007. Basel has an area, as of 2009 , of 23.91 square kilometers (9.23 sq mi). Of this area, 0.95 km (0.37 sq mi) or 4.0% is used for agricultural purposes, while 0.88 km (0.34 sq mi) or 3.7%

5712-521: The Zoo Basel , opened its doors in the south of the city towards Binningen . In 1897 the first World Zionist Congress was held in Basel. Altogether the World Zionist Congress was held in Basel ten times, more than in any other city in the world. On 16 November 1938, the psychedelic drug LSD was first synthesized by Swiss chemist Albert Hofmann at Sandoz Laboratories in Basel. In 1967,

5831-451: 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

5950-514: 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

6069-415: The logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694. In 1695, he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced. Bernoulli's most original work was Ars Conjectandi , published in Basel in 1713, eight years after his death. The work

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6188-600: 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

6307-471: 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

6426-672: 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

6545-534: The 4th century, but were repelled; one such event was the Battle of Solicinium (368). However, in the great invasion of AD 406, the Alemanni appear to have crossed the Rhine a final time, conquering and then settling what is today Alsace and a large part of the Swiss Plateau . The Duchy of Alemannia fell under Frankish rule in the 6th century. The Alemannic and Frankish settlement of Basel gradually grew around

6664-502: The 4th or 5th century and cannot be adduced for the 3rd-century attestation of Basilia . By popular etymology, or simple assonance, the basilisk becomes closely associated with the city, used as heraldic supporter from 1448, represented on coins minted by the city, and frequently found in ornaments. The Middle French form Basle was adopted into English, but this form has fallen gradually out of use although it continues to be used in some sections of British English including

6783-431: The Basel diocese, depicted in bishops' seals of the late medieval period. It is represented in a heraldic context in the early 14th century, not yet as a heraldic charge but as a kind of heraldic achievement flanked by the heraldic shields of the bishop. The staff is also represented in the bishops's seals of the period. The use of the Baselstab in black as the coat of arms of the city was introduced in 1385. From this time,

6902-634: The Ecclesiastical Princes of the Imperial Diet . In 1500 the construction of the Basel Münster was finished. The city had remained neutral through the Swabian War of 1499 despite being plundered by soldiers on both sides. The Treaty of Basel ended the war and granted the Swiss confederates exemptions from the emperor Maximillian's taxes and jurisdictions, separating Switzerland de facto from

7021-535: 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

7140-493: 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} ,

7259-454: 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}}} ,

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7378-626: The Holy Roman Empire. On 9 June 1501, Basel joined the Swiss Confederation as its eleventh canton . It was the only canton that was asked to join, not the other way round. Basel had a strategic location, good relations with Strasbourg and Mulhouse , and control of the corn imports from Alsace, whereas the Swiss lands were becoming overpopulated and had few resources. A provision of the Charter accepting Basel required that in conflicts among

7497-677: 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

7616-706: The Renaissance. Roman control over the area deteriorated in the 3rd century, and Basel became an outpost of the Provincia Maxima Sequanorum formed by Diocletian . Basilia is first named by the Ammianus Marcellinus in his Res Gestae as part of the Roman military fortifications along the Rhine in the late 4th century. The Germanic confederation of the Alemanni attempted to cross the Rhine several times in

7735-917: 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

7854-665: 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

7973-534: 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

8092-426: The ancient bishopric of Augusta Raurica , was established in the 8th century. Under bishop Haito (r. 806–823), the first cathedral was built on the site of the Roman castle (replaced by a Romanesque structure consecrated in 1019). At the partition of the Carolingian Empire through the Treaty of Verdun in 843, Basel was first given to West Francia and became its German exclave. It passed to East Francia with

8211-671: The area while parks, green belts and sports fields made up 8.9%. Out of the forested land, all of the forested land area is covered with heavy forests. Of the agricultural land, 2.5% is used for growing crops and 1.3% is pastures. All the water in the municipality is flowing water. Under the Köppen system , Basel features an oceanic climate (Köppen: Cfb ), although with notable continental influences due to its relatively far inland position with cool to cold, overcast winters and warm to hot, humid summers. The city averages 118.2 days of rain or snow annually and on average receives 842 mm (33.1 in) of precipitation . The wettest month

8330-449: The atmosphere of collaboration between the two brothers turned into rivalry as Johann's own mathematical genius began to mature, with both of them attacking each other in print, and posing difficult mathematical challenges to test each other's skills. By 1697, the relationship had completely broken down. The lunar crater Bernoulli is also named after him jointly with his brother Johann. Jacob Bernoulli's first important contributions were

8449-439: 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

8568-573: 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

8687-507: 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

8806-399: The bishops and the burghers began in the mid-13th century and continued throughout the 14th century. By the late 14th century, the city was for all practical purposes independent although it continued to nominally pledge fealty to the bishops. The House of Habsburg attempted to gain control over the city. This was not successful, but it caused a political split among the burghers of Basel into

8925-451: The city Basel itself. The political structure and agencies of the city and the canton are identical. 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

9044-489: The city municipality limits. The official language of Basel is Swiss Standard German and the main spoken language is the local Basel German dialect. Basel is commonly considered to be the cultural capital of Switzerland and the city is famous for its many museums , including the Kunstmuseum , which is the first collection of art accessible to the public in the world (1661) and the largest museum of art in Switzerland ,

9163-558: The city's centuries-long commitment to humanism , have made Basel a safe haven at times of political unrest in other parts of Europe for such notable people as Erasmus of Rotterdam , the Holbein family, Friedrich Nietzsche , Carl Jung , and in the 20th century also Hermann Hesse and Karl Jaspers . Basel was the seat of a Prince-Bishopric starting in the 11th century, and joined the Swiss Confederacy in 1501. The city has been

9282-423: 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

9401-498: 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

9520-478: The construction of the cathedral of Basel (known locally as the Münster ) began under Henry II, Holy Roman Emperor . In the 11th to 12th century, Basel gradually acquired the characteristics of a medieval city . The main market place is first mentioned in 1091. The first city walls were constructed around 1100 (with improvements made in the mid-13th and in the late 14th century). A city council of nobles and burghers

9639-537: 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

9758-558: The differential calculus in " Nova Methodus pro Maximis et Minimis " published in Acta Eruditorum . They also studied the publications of von Tschirnhaus . It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were among the first to try to understand and apply Leibniz's theories. Jacob collaborated with his brother on various applications of calculus. However

9877-586: The early La Tène period (5th century BC). In the 2nd century BC, there was a village of the Raurici at the site of Basel-Gasfabrik (to the northwest of the Old City, and likely identical with the town of Arialbinnum that was mentioned on the Tabula Peutingeriana ). The unfortified settlement was abandoned in the 1st century BC in favour of an oppidum on the site of Basel Minster , probably in reaction to

9996-451: 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

10115-481: The fundamental result that ∑ 1 n {\displaystyle \sum {\frac {1}{n}}} diverges, which Bernoulli believed were new but they had actually been proved by Pietro Mengoli 40 years earlier and was proved by Nicole Oresme in the 14th century already. Bernoulli could not find a closed form for ∑ 1 n 2 {\displaystyle \sum {\frac {1}{n^{2}}}} , but he did show that it converged to

10234-643: The influential 16th-century martial arts text Kunst des Fechten ("The Art of Fencing"), came from Basel. In 1661 the Amerbaschsches Kabinett , a vast collection of exotic artifacts, coins, medals and books was purchased by Basel. It was to become to the first public museum of art. Its collection became the core of the later Basel Museum of Art . The Bernoulli family, which included important 17th- and 18th-century mathematicians such as Jakob Bernoulli , Johann Bernoulli and Daniel Bernoulli , were from Basel. The 18th-century mathematician Leonhard Euler

10353-414: 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,

10472-423: The killing of a number of men of Leopold III, Duke of Austria . This was seen as a serious breach of the peace , and the city council blamed "foreign ruffians" for this and executed twelve alleged perpetrators. Leopold nevertheless had the city placed under imperial ban , and in a treaty of 9 July, Basel was given a heavy fine and was placed under Habsburg control. To free itself from Habsburg hegemony, Basel joined

10591-575: The land around Farnsburg became a part of Basel. The Schwabe publishing house was founded in 1488 by Johannes Petri and is the oldest publishing house still in business. Johann Froben also operated his printing house in Basel and was notable for publishing works by Erasmus. In 1495, Basel was incorporated into the Upper Rhenish Imperial Circle ; the Bishop of Basel was added to the Bench of

10710-406: The last part of the book, Bernoulli sketches many areas of mathematical probability , including probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; and

10829-417: The law of large numbers. Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity. In 1683, Bernoulli discovered the constant e by studying a question about compound interest which required him to find the value of the following expression (which

10948-1007: 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

11067-454: 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

11186-514: 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

11305-450: 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

11424-516: 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,

11543-518: The old Roman castle in the 6th and 7th century. It appears that Basel surpassed the ancient regional capital of Augusta Raurica by the 7th century; based on the evidence of a gold tremissis (a small gold coin with the value of a third of a solidus ) with the inscription Basilia fit , Basel seems to have minted its own coins in the 7th century. Basel at this time was part of the Archdiocese of Besançon . A separate bishopric of Basel , replacing

11662-445: The other cantons it was to stay neutral and offer its services for mediation. In 1503, the new bishop Christoph von Utenheim refused to give Basel a new constitution; whereupon, to show its power, the city began to build a new city hall. In 1529, the city became Protestant under Oecolampadius and the bishop's seat was moved to Porrentruy . The bishop's crook was however retained as the city's coat of arms. For centuries to come,

11781-416: The population of Basel voted in favor of buying three works of art by painter Pablo Picasso which were at risk of being sold and taken out of the local museum of art, due to a financial crisis on the part of the owner's family. Therefore, Basel became the first city in the world where the population of a political community democratically decided to acquire works of art for a public institution. Pablo Picasso

11900-411: 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

12019-504: 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

12138-452: 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

12257-472: 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

12376-405: The term integral appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called the Bernoulli differential equation , Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola ,

12495-455: The work of Johannes Hudde , Robert Boyle , and Robert Hooke . During this time he also produced an incorrect theory of comets . Bernoulli returned to Switzerland, and began teaching mechanics at the University of Basel from 1683. His doctoral dissertation Solutionem tergemini problematis was submitted in 1684. It appeared in print in 1687. In 1684, Bernoulli married Judith Stupanus; they had two children. During this decade, he also began

12614-524: 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

12733-491: 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

12852-569: 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,

12971-439: Was between 1684 and 1689 that many of the results that were to make up Ars Conjectandi were discovered. People believe he was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on

13090-550: Was born in Basel and studied under Johann Bernoulli. In 1792, the Republic of Rauracia , a revolutionary French client republic , was created. It lasted until 1793. After three years of political agitation and a short civil war in 1833 the disadvantaged countryside seceded from the Canton of Basel, forming the half canton of Basel-Landschaft . Between 1861 and 1878 the city walls were slighted . On 3 July 1874, Switzerland's first zoo,

13209-427: 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

13328-451: 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

13447-400: 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

13566-507: 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

13685-605: Was engraved rather than a logarithmic one. Translation of Latin inscription: Basel Basel ( / ˈ b ɑː z əl / BAH -zəl ; German: [ˈbaːzl̩] ), also known as Basle , is a city in northwestern Switzerland on the River Rhine (at the transition from the High to the Upper Rhine ). Basel is Switzerland's third-most-populous city (after Zurich and Geneva ), with 177,595 inhabitants within

13804-429: Was established. Basel became the focal point of western Christendom during the 15th century Council of Basel (1431–1449), including the 1439 election of antipope Felix V . In 1459, Pope Pius II endowed the University of Basel , where such notables as Erasmus of Rotterdam and Paracelsus later taught. At the same time the new craft of printing was introduced to Basel by apprentices of Johann Gutenberg . In 1461,

13923-571: 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

14042-540: Was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability. The book also covers other related subjects, including a review of combinatorics , in particular the work of van Schooten, Leibniz, and Prestet, as well as the use of Bernoulli numbers in a discussion of the exponential series. Inspired by Huygens' work, Bernoulli also gives many examples on how much one would expect to win playing various games of chance. The term Bernoulli trial resulted from this work. In

14161-607: Was so moved by the gesture that he subsequently gifted the city with an additional three paintings. Basel has often been the site of peace negotiations and other international meetings. The Treaty of Basel (1499) ended the Swabian War . Two years later Basel joined the Swiss Confederation . The Peace of Basel in 1795 between the French Republic and Prussia and Spain ended the First Coalition against France during

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