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96-621: 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 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

192-585: 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 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

288-517: 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

384-415: 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 was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways,

480-570: 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 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

576-545: A growing economic discrepancy. The Catholic, predominantly rural central-Swiss cantons were surrounded by Protestant cantons with increasingly commercial economies. The politically dominant cantons were Zürich and Bern (both Protestant), but the Catholic cantons were influential since the Second War of Kappel in 1531. A 1655 attempt (led by Zürich) to restructure the federation was blocked by Catholic opposition, which led to

672-792: 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 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

768-653: 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 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

864-469: 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

960-402: 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 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

1056-459: 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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1152-460: A solution to several unsolved problems in number theory and analysis, including 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

1248-643: 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 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

1344-501: 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

1440-758: A whole, expanded through military conquest: the Aargau was conquered in 1415 and the Thurgau in 1460. In both cases, the Swiss profited from weakness in the Habsburg dukes. In the south, Uri led a military territorial expansion that (after many setbacks) would by 1515 lead to the conquest of the Ticino .None of these territories became members of the confederacy; they had the status of condominiums (regions administered by several cantons). The reason for these Swiss victories

1536-522: A younger brother, Johann Heinrich. Soon after 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

1632-404: 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 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

1728-624: Is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain 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

1824-493: Is the precursor of the modern state of Switzerland . It formed at the end of the 13th century, from a nucleus in what is now Central Switzerland , expanding to include the cities of Zürich and Bern by the middle of the 14th century. This formed a rare union of rural and urban communes , all of which enjoyed imperial immediacy in the Holy Roman Empire. This confederation of eight cantons ( Acht Orte )

1920-429: 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

2016-757: 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

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2112-604: The Tagsatzung , developed during the 15th century. Pacts and renewals (or modernizations) of earlier alliances reinforced the confederacy. The individual interests of the cantons clashed in the Old Zürich War (1436–1450), caused by territorial conflict among Zürich and the central Swiss cantons over the succession of the Count of Toggenburg . Although Zürich entered an alliance with the Habsburg dukes, it then rejoined

2208-624: The Tagsatzung in the Stanser Verkommnis restored order and assuaged the rural cantons' complaints, with Fribourg and Solothurn accepted into the confederation. While the treaty restricted freedom of assembly (many skirmishes arose from unauthorised expeditions by soldiers from the Burgundian Wars), it reinforced agreements amongst the cantons in the earlier Sempacherbrief and Pfaffenbrief . The civil war during

2304-795: The Freie Ämter ("Free Districts"), the Thurgau , the Rhine valley, and Sargans , and furthermore the Catholic cantons were excluded from the administration of the County of Baden . The "German bailiwicks" ( German : Deutsche Gemeine Vogteien, Gemeine Herrschaften ) were generally governed by the Acht Orte apart from Bern until 1712, when Bern joined the sovereign powers: Several bailiwicks ( Vogteien ) were generally referred to as "transmontane bailiwicks" ( German : Ennetbergische Vogteien , Italian : Baliaggi Ultramontani ). In 1440, Uri conquered

2400-558: 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 the University of Basel to succeed the recently deceased Johann Bernoulli. In 1753 he bought

2496-1037: 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

2592-459: 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

2688-590: The Euler approximations . The most notable of these approximations are Euler's method and the Euler–Maclaurin formula . Old Swiss Confederacy The Old Swiss Confederacy , also known as Switzerland or the Swiss Confederacy , was a loose confederation of independent small states ( cantons , German Orte or Stände ), initially within the Holy Roman Empire . It

2784-820: The Gotthard Pass went through the area. This pass was important in the Hohenstaufen struggle against the Italian Lombard League . Due to the fragmentation of Swabia following the collapse of the Hohenstaufens, many nations began to compete for land. The Habsburgs in particular were a major threat to the Swiss states. King Rudolf I added large amounts of territory in Switzerland and Swabia to his domain , and sieged down Bern in 1289 to enforce imperial taxes. This aggression from

2880-546: 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 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

2976-745: The Leventina Valley from the Visconti , dukes of Milan . Some of this territory had previously been annexed between 1403 and 1422. Further territories were acquired in 1500; see History of Ticino for further details . Three bailiwicks, all now in the Ticino , were condominiums of the Forest cantons of Uri, Schwyz and Nidwalden: Four other Ticinese bailiwicks were condominiums of the Zwölf Orte (the original 13 cantons, minus Appenzell) from 1512: Another three bailiwicks were condominiums of

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3072-518: 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, 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

3168-522: The Zwölf Orte from 1512, but were lost from the Confederacy three years later and are all now comuni of Lombardy or Piedmont : Some territories were separate subjects of cantons or associates, Einzelörtische Untertanen von Länderorten und Zugewandten : Delta (letter) A river delta (originally, the delta of the Nile River ) is so named because its shape approximates

3264-593: The first battle of Villmergen in 1656; the Catholic party won, cementing the status quo . The problems remained unsolved, erupting again in 1712 with the second battle of Villmergen . This time the Protestant cantons won, dominating the confederation. True reform, however, was impossible; the individual interests of the thirteen members were too diverse, and the absolutist cantonal governments resisted all attempts at confederation-wide administration. Foreign policy remained fragmented. The (Alte) Eidgenossenschaft

3360-470: The imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , 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

3456-524: 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

3552-618: The second war of Villmergen ). The Catholic cantons were excluded from administering the condominiums in the Aargau, the Thurgau and the Rhine valley; in their place, Bern became co-sovereign of these regions. The confederation expanded in several stages: first to the Eight Cantons ( Acht Orte ), then in 1481 to ten, in 1501 to twelve, and finally to thirteen cantons ( Dreizehn Orte ). Associates (Zugewandte Orte) were close allies of

3648-399: The 16th century Zürich permanently assumed the chair ( Vorort ) and Baden became the seat. The Tagsatzung dealt with inter-cantonal affairs and was the court of last resort in disputes between member states, imposing sanctions on dissenting members. It also administered the condominiums; the reeves were delegated for two years, each time by a different canton. A unifying treaty of

3744-524: The Catholic and Protestant factions, but the confederacy as a whole continued to exist. A common foreign policy was blocked, however, by the impasse. During the Thirty Years' War , religious disagreements among the cantons kept the confederacy neutral and spared it from belligerents. At the Peace of Westphalia , the Swiss delegation was granted formal recognition of the confederacy as a state independent of

3840-483: The Confederacy managed to stop the growing Burgundian threat. In the Swabian War against Holy Roman Emperor Maximilian I , the Swiss were victorious and exempted from imperial legislation. The associated cities of Basel and Schaffhausen joined the confederacy as a result of that conflict, and Appenzell followed suit in 1513 as the thirteenth member. The federation of thirteen cantons ( Dreizehn Orte ) constituted

3936-495: The Empire, and they acknowledged the authority of the Holy Roman Empire. The initial pact was augmented by pacts with the cities of Lucerne , Zürich , and Bern . This union of rural and urban communes, which enjoyed the status of imperial immediacy within the Holy Roman Empire , was engendered by pressure from Habsburg dukes and kings who had ruled much of the land. Bern in particular had fought against local nobles, including

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4032-611: The English Switzerland beginning during the mid-16th century. From that time the Confederacy was seen as a single state, also known as the Swiss Republic ( Republic der Schweitzer , République des Suisses and Republica Helvetiorum by Josias Simmler in 1576) after the fashion of calling individual urban cantons republics (such as the Republics of Zürich , Berne and Basel ). The nucleus of

4128-539: 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

4224-671: The Habsburgs would likely be a major factor in the creation of the Swiss Confederacy. The foundation of the Confederacy is marked by the Rütlischwur (dated to 1307 by Aegidius Tschudi ) or the 1315 Pact of Brunnen . Since 1889, the Federal Charter of 1291 among the rural communes of Uri , Schwyz , and Unterwalden has been considered the founding document of the confederacy. The early Swiss Confederacy

4320-422: The Habsburgs, on several occasions, and as such was eager to join the Confederacy. In several battles with Habsburg armies, the Swiss were victorious; they conquered the rural areas of Glarus and Zug , which became members of the confederacy. This expansion greatly changed the social climate in the Confederacy. In the original 3 cantons, citizens all held equal rights, but in the newly acquired urban cities, power

4416-676: The Holy Roman Empire. Growing social differences and an increasing absolutism in the city cantons during the Ancien Régime led to local popular revolts . An uprising during the post-war depression after the Thirty Years' War escalated to the Swiss peasant war of 1653 in Lucerne, Bern, Basel, Solothurn and the Aargau. The revolt was put down swiftly by force and with the help of many cantons. Religious differences were accentuated by

4512-530: The Old Swiss Confederacy until its demise in 1798. The expansion of the confederacy was stopped by the Swiss defeat in the 1515 Battle of Marignano . Only Bern and Fribourg were still able to conquer the Vaud in 1536; the latter primarily became part of the canton of Bern , with a small portion under the jurisdiction of Fribourg. The Reformation in Switzerland led to doctrinal division amongst

4608-497: The Old Swiss Confederacy was an alliance among the valley communities of the central Alps to facilitate management of common interests (such as trade) and ensure peace along trade routes through the mountains. Previously, the communities, also known collectively as the "Forest States", had been granted imperial immediacy and autonomy during the reign of the Hohenstaufens . The region was endowed with these privileges because

4704-573: The Old Swiss Confederacy was the Stanser Verkommnis of 1481. Conflicts between rural and urban cantons and disagreements over the bounty of the Burgundian Wars had led to skirmishes. The urban cantons of Bern, Zurich, and Luzern in particular wanted to keep a large share of the bounty. The city-states of Fribourg and Solothurn wanted to join the confederacy, but were mistrusted by the central Swiss rural cantons. The compromise by

4800-400: The Old Swiss Confederacy, connected to the union by alliance treaties with all or some of the individual members of the confederacy. Three of the associates were known as Engere Zugewandte : Two federations were known as Ewige Mitverbündete : There were two Evangelische Zugewandte : Condominiums ( German : Gemeine Herrschaften ) were common subject territories under

4896-488: The Reformation ended in a stalemate. The Catholic cantons could block council decisions but, due to geographic and economic factors, could not prevail over the Protestant cantons. Both factions began to hold separate councils, still meeting at a common Tagsatzung (although the common council was deadlocked by disagreements between both factions until 1712, when the Protestant cantons gained power after their victory in

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4992-544: The abbot and the city of St. Gallen , Biel , Rottweil , Mulhouse and others. These allies (known as the Zugewandte Orte ) became closely associated with the confederacy, but were not accepted as full members. They would be known as Swiss Associates . The Burgundian Wars prompted a further enlargement of the confederacy; Fribourg and Solothurn were accepted in 1481. By defeating the Duchy of Burgundy in this war,

5088-399: 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 the years, Euler entered this competition 15 times, winning 12 of them. Johann Bernoulli's two sons, Daniel and Nicolaus , entered into service at

5184-413: The administration of several cantons. They were governed by reeves ( Vögte ) delegated for two years, each time from another of the responsible cantons. Bern initially did not participate in the administration of some of the eastern condominiums, as it had no part in their conquest and its interests were focused more on the western border. In 1712, Bern replaced the Catholic cantons in the administration of

5280-536: The age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in 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

5376-483: The angles as capital letters. He gave the current definition of the constant e {\displaystyle e} , 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

5472-540: The ascension of the twelve-year-old Peter II . The nobility, suspicious of 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

5568-671: The cantons. Zürich, Bern, Basel, Schaffhausen and associates Biel, Mulhouse, Neuchâtel, Geneva and the city of St. Gallen became Protestant ; other members of the confederation and the Valais remained Catholic . In Glarus, Appenzell, in the Grisons and in most condominiums both religions coexisted; Appenzell split in 1597 into a Catholic Appenzell Innerrhoden and a Protestant Appenzell Ausserrhoden . The division led to civil war (the Wars of Kappel ) and separate alliances with foreign powers by

5664-417: The capital of Prussia . Euler is credited for popularizing 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

5760-419: The confederacy, the cantons for the first time used the term Eidgenossenschaft . The first treaty uniting the eight members of the confederacy was the Sempacherbrief  [ de ; fr ] of 1393, concluded after victories over the Habsburgs at Sempach in 1386 and Näfels in 1388, which forbade a member from unilaterally beginning a war without the consent of the other cantons. A federal diet ,

5856-406: The confederacy. The confederation had become so close a political alliance that it no longer tolerated separatist tendencies in its members. The Tagsatzung was the confederation council, typically meeting several times a year. Each canton delegated two representatives (including the associate states, which had no vote). The canton where the delegates met initially chaired the gathering, but during

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5952-461: The confederates into Reformed and Catholic parties, resulting in internal conflict from the 16th to the 18th centuries; as a result, the federal diet ( Tagsatzung ) was often paralysed by hostility between the factions. The Swiss Confederacy fell to a French invasion in 1798, after which it became the short-lived Helvetic Republic . The adjective "old" was introduced after the Napoleonic era with Ancien Régime , retronyms distinguishing

6048-498: 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

6144-402: 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 the Introductio in analysin infinitorum was published and in 1755 a text on differential calculus called

6240-419: 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 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

6336-533: 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

6432-424: 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 the cartography he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains

6528-610: 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 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

6624-413: 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,

6720-460: 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 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

6816-481: 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 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

6912-501: The motion of rigid bodies . He also made substantial contributions to 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

7008-454: 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

7104-539: The notion of a mathematical function . He is also known for his work in mechanics , fluid dynamics , optics , astronomy , and music theory . Euler 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

7200-605: 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 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

7296-402: 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 a water jet in my garden: Euler calculated

7392-410: The pre-Napoleonic from the restored confederation. During its existence the confederacy was known as Eidgenossenschaft or Eydtgnoschafft ("oath fellowship"), in reference to treaties among cantons; this term was first used in the 1370 Pfaffenbrief . Territories of the confederacy came to be known collectively as Schweiz or Schweizerland ( Schwytzerland in contemporary spelling), with

7488-516: The ranks in the academy and was made a professor of physics in 1731. He also left 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

7584-503: 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

7680-674: 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 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

7776-477: 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 was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in

7872-547: The threat of the Thirty Years' War ), although many Swiss served privately as mercenaries in the Italian Wars and during the early modern period. After the Swabian War of 1499 the confederacy was a de facto independent state throughout the early modern period , although still nominally part of the Holy Roman Empire until 1648 when the Treaty of Westphalia ended the Thirty Years' War . The Swiss Reformation divided

7968-436: The triangular uppercase letter delta. Contrary to a popular legend, this use of the word delta was not coined by Herodotus . In Ancient Greek , delta represented a voiced dental plosive IPA: [d] . In Modern Greek , it represents a voiced dental fricative IPA: [ð] , like the "th" in "that" or "this" (while IPA: [d] in foreign words is instead commonly transcribed as ντ). Delta

8064-464: 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, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he

8160-498: 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 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

8256-548: 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 the Neva River . Of their thirteen children, only five survived childhood, three sons and two daughters. Their first son

8352-524: 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 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

8448-802: 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 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

8544-423: 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

8640-420: 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 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

8736-596: Was in the hands of the wealthy Burgomeisters . This led into a shift away from the traditional Swiss egalitarianism enjoyed by citizens in the original Confederacy. From 1353 to 1481, the federation of eight cantons —known in German as the Acht Orte (Eight Cantons)—consolidated its position. The members (especially the cities) enlarged their territory at the expense of local counts—primarily by buying judicial rights , but sometimes by force. The Eidgenossenschaft , as

8832-520: Was initially united not by a single pact, but by overlapping pacts and bilateral treaties between members. The parties generally agreed to preserve the peace, aid in military endeavours and arbitrate disputes. Slowly, the members began to see the confederation as a unifying entity. In the Pfaffenbrief , a treaty of 1370 among six of the eight members (Glarus and Bern did not participate) forbidding feuds and denying clerical courts jurisdiction over

8928-532: Was politically and militarily successful for more than a century, culminating in the Burgundy Wars of the 1470s which established it as a power in the complicated political landscape dominated by France and the Habsburgs . Its success resulted in the addition of more confederates, increasing the number of cantons to thirteen ( Dreizehn Orte ) by 1513. The confederacy pledged neutrality in 1647 (under

9024-432: Was simply a defensive pact, but over time the states grew closer and closer. Following a victory against the Habsburgs in 1315, the members of the Confederacy vowed not to form alliances with outside states without consent of the rest of the Confederacy. They also agreed to resolve all disputes peacefully and to support one another in both external and internal affairs. At this point however, the Swiss were still subordinate to

9120-510: Was their innovative military tactics. Their perfection of the Pike Square made them excellent defensive warriors in their home mountain terrain, and they became highly sought after mercenaries throughout Europe (ex Swiss Guard ). At this time, the eight cantons gradually increased their influence on neighbouring cities and regions through additional alliances. Individual cantons concluded pacts with Fribourg , Appenzell , Schaffhausen ,

9216-543: 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 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

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