Leonhard Euler (/ˈɔɪlər/OY-lər;[b] 15 April 1707 – 18 September 1783) was a Swisspolymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies ofgraph theory andtopology and made influential discoveries in many other branches of mathematics, such asanalytic number theory,complex analysis, andinfinitesimal calculus. He also introduced much of modern mathematical terminology andnotation, including the notion of amathematical function.[3] He is known for his work inmechanics,fluid dynamics, optics, astronomy, and music theory.[4] Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory".[5] He spent most of his adult life inSaint Petersburg, Russia, and inBerlin, then the capital ofPrussia.
Euler is credited for popularizing the Greek letter (lowercasepi) to denotethe ratio of a circle's circumference to its diameter, as well as first using the notation for the value of a function, the letter to express theimaginary unit, the Greek letter (capitalsigma) to expresssummations, the Greek letter (capitaldelta) forfinite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters.[6] He gave the current definition of the constant, the base of thenatural logarithm, now known asEuler's number.[7] Euler made contributions toapplied mathematics andengineering, such as his study of ships, which helped navigation; his three volumes on optics, which contributed to the design ofmicroscopes andtelescopes; and his studies of beam bending and column critical loads.[8]
Euler is credited with being the first to developgraph theory (partly as a solution for the problem of theSeven Bridges of Königsberg, which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, solving several unsolved problems in number theory and analysis, including the famousBasel problem. Euler has also been credited for discovering that the sum of the numbers of vertices and faces minus the number of edges of apolyhedron that has no holes equals 2, a number now commonly known as theEuler characteristic. In physics, Euler reformulatedIsaac Newton'slaws of motion intonew laws in his two-volume workMechanica to better explain the motion ofrigid bodies. He contributed to the study ofelastic deformations of solid objects. Euler formulated thepartial differential equations for the motion ofinviscid fluid,[8] and laid the mathematical foundations ofpotential theory.[5]
Euler is regarded as arguably the most prolific contributor in the history of mathematics and science, and the greatest mathematician of the 18th century.[9][8] His 866 publications and his correspondence are being collected in theOpera Omnia Leonhard Euler.[10][11][12] Several great mathematicians who worked after Euler's death have recognised his importance in the field:Pierre-Simon Laplace said, "Read Euler, read Euler, he is the master of us all";[13][c]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."[14][d]
Leonhard Euler was born inBasel on 15 April 1707 to Paul III Euler, a pastor of theReformed Church, and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics.[16] He was the oldest of four children, with two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich.[17][16] Soon after Leonhard's birth, the Eulers moved from Basel toRiehen, Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood.[16]
From a young age, Euler received schooling in mathematics from his father, who had taken courses fromJacob Bernoulli some years earlier at theUniversity of Basel. Around 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.[16]
In 1720, at age 13, Euler enrolled at theUniversity of Basel.[4] Attending university at such a young age was not unusual at the time.[16] The course on elementary mathematics was given byJohann 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:[18]
the famous professor Johann Bernoulli [...] made it a special pleasure for himself to help me along in the mathematical sciences. Private lessons, however, he refused because of his busy schedule. However, he gave me a far more salutary advice, which consisted in myself getting a hold of some of the more difficult mathematical books and working through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when he resolved one of my objections, ten others at once disappeared, which certainly is the best method of making happy progress in the mathematical sciences.
During this time, Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor.[19][20]
In 1723, Euler received aMaster of Philosophy with a dissertation that compared the philosophies ofRené Descartes andIsaac Newton.[16] Afterwards, he enrolled in the theological faculty of the University of Basel.[20]
In 1726, Euler completed a dissertation on thepropagation of sound titledDe Sono,[21][22] with which he unsuccessfully attempted to obtain a position at the University of Basel.[23] In 1727, he entered theParis Academy prize competition (offered annually and later biennially by the academy beginning in 1720)[24] for the first time. The problem posed that year was to find the best way to place themasts on a ship.Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place.[25] Over the years, Euler entered this competition 15 times,[24] winning 12 of them.[25]
1957Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Johann Bernoulli's two sons,Daniel andNicolaus, entered into service at theImperial Russian Academy of Sciences inSaint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available.[23] On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia.[26][27] 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.[23] 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.[23]
Euler arrived in Saint Petersburg in May 1727.[23][20] 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.[28] Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as a medic in theRussian Navy.[29]
The academy at Saint Petersburg, established byPeter 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.[25] The academy's benefactress,Catherine I, who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg.[30] The Russian conservative nobility then gained power upon the ascension of the twelve-year-oldPeter II.[30] 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.[30]
Conditions improved slightly after the death of Peter II in 1730 and the German-influencedAnna of Russia assumed power.[31] Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731.[31] He also left the Russian Navy, refusing a promotion tolieutenant.[31] 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.[32] In January 1734, he married Katharina Gsell (1707–1773), a daughter ofGeorg Gsell.[33]Frederick II had made an attempt to recruit the services of Euler for his newly establishedBerlin Academy in 1740, but Euler initially preferred to stay in St Petersburg.[34] But after Empress Anna died and Frederick II agreed to pay 1600ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave for Berlin, arguing he was in need of a milder climate for his eyesight.[34] The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members.[34]
Euler became the tutor forFriederike Charlotte of Brandenburg-Schwedt, the Princess ofAnhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitledLetters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess.[44] 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 theLetters testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.[37]
Despite Euler's immense contribution to the academy's prestige and having been put forward as a candidate for its presidency byJean le Rond d'Alembert,Frederick II named himself as its president.[43] 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, the polar opposite ofVoltaire, 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.[37] Frederick also expressed disappointment with Euler's practical engineering abilities, stating:
I wanted to have 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 inSanssouci. 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![45]
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.[46]
Throughout his stay in Berlin, Euler maintained a strong connection to the academy in St. Petersburg and also published 109 papers in Russia.[47] He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin.[47] In 1760, with theSeven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops.[42] Upon learning of this event,General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, withEmpress Elizabeth of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time.[48] Euler decided to leave Berlin in 1766 and return to Russia.[49]
During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published.[50] This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to theSt. Petersburg Academy, which had retained him as a member and paid him an annual stipend. Euler'sIntroductio 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.[51] He was even involved in the design of the water fountains atSanssouci, the King's summer palace.[52]
The political situation in Russia stabilized afterCatherine 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 was assisted by his studentAnders Johan Lexell.[53] While living in St. Petersburg, a fire in 1771 destroyed his home.[54]
On 7 January 1734, Euler married Katharina Gsell, daughter ofGeorg Gsell, a painter at the Academy Gymnasium in Saint Petersburg.[33] The couple bought a house by theNeva River.
Three years after his wife's death in 1773,[54] Euler married her half-sister, Salome Abigail Gsell.[57] 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.[34]
Early in his life, Euler memorizedVirgil'sAeneid, and by old age, he could recite the poem and give the first and last sentence on each page of the edition from which he had learnt it.[58][59] Euler knew the first hundred prime numbers and could give each of their powers up to the sixth degree.[60]
Euler was known as a generous and kind person, not neurotic as seen in some geniuses, keeping his good-natured disposition even after becoming entirely blind.[60]
Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly dying of fever,[61] he became almost blind in his right eye. Euler blamed thecartography he performed for the St. Petersburg Academy for his condition,[62] but the cause of his blindness remains the subject of speculation.[63][64] Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick called him "Cyclops". Euler said of his loss of vision, "Now I will have fewer distractions."[62] In 1766 acataract in his left eye was discovered. Thoughcouching of the cataract temporarily improved his vision, complications rendered him almost totally blind in the left eye as well.[39] 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;[65] in 1775, he produced, on average, one mathematical paper per week.[39]
Euler worked in almost all areas of mathematics, includinggeometry,infinitesimal calculus,trigonometry,algebra, andnumber theory, as well ascontinuum physics,lunar theory, and other areas ofphysics. 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 80quarto volumes.[39] Euler's name is associated with alarge 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, while other researchers credit Euler for a third of the output in mathematics in that century.[6]
Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of afunction[3] and was the first to writef(x) to denote the functionf applied to the argumentx. He also introduced the modern notation for thetrigonometric functions, the lettere for the base of thenatural logarithm (now also known asEuler's number), the Greek letterΣ for summations and the letteri to denote theimaginary unit.[68] The use of the Greek letterπ to denote theratio of a circle's circumference to its diameter was also popularized by Euler, although it originated withWelsh mathematicianWilliam Jones.[69]
The development ofinfinitesimal calculus was at the forefront of 18th-century mathematical research, and theBernoullis—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 ofmathematical rigour[70] (in particular his reliance on the principle of thegenerality of algebra), his ideas led to many great advances.Euler is well known inanalysis for his frequent use and development ofpower series, the expression of functions as sums of infinitely many terms,[71] such as
Euler's use of power series enabled him to solve theBasel 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 byPietro Mengoli in 1644, and by the 1730s was a famous open problem, popularized byJacob Bernoulli and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that:[72][73][70]
Euler introduced the use of theexponential function andlogarithms inanalytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative andcomplex numbers, thus greatly expanding the scope of mathematical applications of logarithms.[68] He also defined the exponential function for complex numbers and discovered its relation to thetrigonometric functions. For anyreal numberφ (taken to be radians),Euler's formula states that thecomplex exponential function satisfies
which was called "the most remarkable formula in mathematics" byRichard Feynman.[75]
A special case of the above formula is known asEuler's identity,
Euler's interest in number theory can be traced to the influence ofChristian Goldbach,[80] his friend in the St. Petersburg Academy.[61] Much of Euler's early work on number theory was based on the work ofPierre 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 (Fermat numbers) are prime.[81]
Map ofKönigsberg in Euler's time showing the actual layout of theseven bridges, highlighting the riverPregel and the bridges
In 1735, Euler presented a solution to the problem known as theSeven Bridges of Königsberg.[88] The city ofKönigsberg,Prussia was set on thePregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once. Euler showed that it is not possible: there is noEulerian path. This solution is considered to be the first theorem ofgraph theory.[88]
Euler also discovered theformula relating the number of vertices, edges, and faces of aconvex polyhedron,[89] and hence of aplanar graph. The constant in this formula is now known as theEuler characteristic for the graph (or other mathematical object), and is related to thegenus of the object.[90] The study and generalization of this formula, specifically byCauchy[91] andL'Huilier,[92] is at the origin oftopology.[89]
Euler helped develop theEuler–Bernoulli beam equation, which became a cornerstone of engineering.[97] Besides successfully applying his analytic tools to problems inclassical mechanics, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multipleParis Academy Prizes over the course of his career. His accomplishments include determining with great accuracy theorbits ofcomets and other celestial bodies, understanding the nature of comets, and calculating theparallax of the Sun. His calculations contributed to the development of accuratelongitude tables.[98]
Influid dynamics, Euler was the first to predict the phenomenon ofcavitation, in 1754, long before its first observation in the late 19th century, and theEuler number used in fluid flow calculations comes from his related work on the efficiency ofturbines.[102] In 1757 he published an important set of equations forinviscid flow influid dynamics, that are now known as theEuler equations.[103]
An Euler diagram is adiagrammatic means of representingsets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depictsets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents theelements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection,subset, anddisjointness). Curves whose interior zones do not intersect representdisjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (theintersection of the sets). A curve that is contained completely within the interior zone of another represents asubset of it.
Euler diagrams (and their refinement toVenn diagrams) were incorporated as part of instruction inset theory as part of thenew math movement in the 1960s.[106] Since then, they have come into wide use as a way of visualizing combinations of characteristics.[107]
In his 1760 paperA General Investigation into the Mortality and Multiplication of the Human Species Euler produced a model which showed how a population with constant fertility and mortality might grow geometrically using a difference equation. Under this geometric growth Euler also examined relationships among various demographic indices showing how they might be used to produce estimates when observations were missing. Three papers published around 150 years later byAlfred J. Lotka (1907, 1911 (with F.R. Sharpe) and 1922) adopted a similar approach to Euler's and produced their Stable Population Model. These marked the start of 20th century formal demographic modelling.[108][109][110][111][112][113]
One of Euler's more unusual interests was the application ofmathematical ideas in music. In 1739 he wrote theTentamen novae theoriae musicae (Attempt at a New Theory of Music), hoping to eventually incorporatemusical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.[114] Even when dealing with music, Euler's approach is mainly mathematical,[115] for instance, his introduction ofbinary logarithms as a way of numerically describing the subdivision ofoctaves into fractional parts.[116] His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that remained with him throughout his life.[115]
A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2mA, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2m (where "m is an indefinite number, small or large, so long as the sounds are perceptible"[117]), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2m.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2m.5, major third + minor sixth (C–E–C); the fourth is 2m.32, two-fourths and a tone (C–F–B♭–C); the fifth is 2m.3.5 (C–E–G–B–C); etc. Genres 12 (2m.33.5), 13 (2m.32.52) and 14 (2m.3.53) are corrected versions of thediatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2m.33.52) is the "diatonico-chromatic", "used generally in all compositions",[118] and which turns out to be identical with the system described byJohann Mattheson.[119] Euler later envisaged the possibility of describing genres including the prime number 7.[120]
Euler devised a specific graph, theSpeculum musicum,[121][122] to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (seeabove). The device drew renewed interest as theTonnetz inNeo-Riemannian theory (see alsoLattice (music)).[123]
Euler further used the principle of the "exponent" to propose a derivation of thegradus suavitatis (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he consideredjust intonation, i.e. 1 and only the prime numbers 3 and 5.[124] Formulas have been proposed extending this system to any number of prime numbers, e.g. in the formwherepi are prime numbers andki their exponents.[125]
Euler was religious throughout his life.[20] Much of what is known of his religious beliefs can be deduced from hisLetters to a German Princess and an earlier work,Rettung der Göttlichen Offenbahrung gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These show that Euler was a devout Christian who believed the Bible to be inspired; theRettung was primarily an argument for thedivine inspiration of scripture.[126][127]
Euler opposed the concepts ofLeibniz'smonadism and the philosophy ofChristian Wolff.[128] He insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler called Wolff's ideas "heathen and atheistic".[129]
There is a legend[130] inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopherDenis Diderot was visiting Russia on Catherine the Great's invitation. The Empress was alarmed that Diderot's arguments for atheism were influencing members of her court, and so Euler was asked to confront him. Diderot was informed that a learned mathematician had produced a proof of theexistence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced thisnon sequitur:
"Sir,, hence God exists –reply!"
Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request Catherine granted. However amusing the anecdote may be, it isapocryphal, given that Diderot himself did research in mathematics.[131]The legend was apparently first told byDieudonné Thiébault with embellishment byAugustus De Morgan.[130]
Euler is widely recognized as one of the greatest mathematicians of all time, and more likely than not the most prolific contributor to mathematics and science.[8] Mathematician and physicistJohn von Neumann called Euler "the greatestvirtuoso of the period".[132] MathematicianFrançois Arago said, "Euler calculated without any apparent effort, just as men breathe and as eagles sustain themselves in air".[133] He is generally ranked right belowCarl Friedrich Gauss,Isaac Newton, andArchimedes among the greatest mathematicians of all time,[133] while some rank him as equal with them.[134] Physicist and mathematicianHenri Poincaré called Euler the "god of mathematics".[135]
French mathematicianAndré Weil noted that Euler stood above his contemporaries and more than anyone else was able to cement himself as the leading force of his era's mathematics:[132]
No mathematician ever attained such a position of undisputed leadership in all branches of mathematics, pure and applied, as Euler did for the best part of the eighteenth century.
Swiss mathematicianNicolas Fuss noted Euler's extraordinary memory and breadth of knowledge, saying:[5]
Knowledge that we call erudition was not inimical to him. He had read all the best Roman writers, knew perfectly the ancient history of mathematics, held in his memory the historical events of all times and peoples, and could without hesitation adduce by way of examples the most trifling of historical events. He knew more about medicine, botany, and chemistry than might be expected of someone who had not worked especially in those sciences.
Dioptrica, published in three volumes beginning in 1769[99]
It took until 1830 for the bulk of Euler's posthumous works to be individually published,[146] with an additional batch of 61 unpublished works discovered byPaul Heinrich von Fuss (Euler's great-grandson andNicolas Fuss's son) and published as a collection in 1862.[146][147] A chronological catalog of Euler's works was compiled by Swedish mathematicianGustaf Eneström and published from 1910 to 1913.[148] The catalog, known as theEneström index, numbers Euler's works from E1 to E866.[149] The Euler Archive was started atDartmouth College[150] before moving to theMathematical Association of America[151] and, most recently, toUniversity of the Pacific in 2017.[152]
In 1907, theSwiss Academy of Sciences created theEuler Commission and charged it with the publication of Euler's complete works. After several delays in the 19th century,[146] the first volume of theOpera Omnia, was published in 1911.[153] However, the discovery of new manuscripts continued to increase the magnitude of this project. Fortunately, the publication of Euler's Opera Omnia has made steady progress, with over 70 volumes (averaging 426 pages each) published by 2006 and 80 volumes published by 2022.[154][11][6] These volumes are organized into four series. The first series compiles the works on analysis, algebra, and number theory; it consists of 29 volumes and numbers over 14,000 pages. The 31 volumes of Series II, amounting to 10,660 pages, contain the works on mechanics, astronomy, and engineering. Series III contains 12 volumes on physics. Series IV, which contains the massive amount of Euler's correspondence, unpublished manuscripts, and notes only began compilation in 1967. After publishing 8 print volumes in Series IV, the project decided in 2022 to publish its remaining projected volumes in Series IV in online format only.[11][153][6]
Illustration fromSolutio problematis... a. 1743 propositi published inActa Eruditorum, 1744
The title page of Euler'sMethodus inveniendi lineas curvas
^The quote appeared inGugliemo Libri's review of a recently published collection of correspondence among eighteenth-century mathematicians: "... nous rappellerions que Laplace lui même, ... ne cessait de répéter aux jeunes mathématiciens ces paroles mémorables que nous avons entendues de sa propre bouche : 'Lisez Euler, lisez Euler, c'est notre maître à tous.'" [... we would recall that Laplace himself, ... never ceased to repeat to young mathematicians these memorable words that we heard from his own mouth: 'Read Euler, read Euler, he is our master in everything.'][155]
^Gauss wrote this in a letter toPaul Fuss dated September 11, 1849:[15] "Die besondere Herausgabe der kleinern Eulerschen Abhandlungen ist gewiß etwas höchst verdienstliches, [...] und das Studium aller Eulerschen Arbeiten doch stets die beste durch nichts anderes zu ersetzende Schule für die verschiedenen mathematischen Gebiete bleiben wird." [The special publication of the smaller Euler treatises is certainly something highly deserving, [...] and the study of all Euler's works will always remain the best school for the various mathematical fields, which cannot be replaced by anything else.]
^Dunham 1999, p. xiii "Lisez Euler, lisez Euler, c'est notre maître à tous."
^Grinstein, Louise; Lipsey, Sally I. (2001). "Euler, Leonhard (1707–1783)".Encyclopedia of Mathematics Education.Routledge. p. 235.ISBN978-0-415-76368-4.
^Euler, Leonhard (1727).Dissertatio physica de sono [Physical dissertation on sound] (in Latin). Basel: E. and J. R. Thurnisiorum. Retrieved2021-06-06 – via Euler archive. Translated into English as Bruce, Ian."Euler's Dissertation De Sono: E002"(PDF).Some Mathematical Works of the 17th & 18th Centuries, including Newton's Principia, Euler's Mechanica, Introductio in Analysin, etc., translated mainly from Latin into English. Retrieved2021-06-12.
^abEves, Howard W. (1969). "Euler's blindness".In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes, Quadrants III and IV. Prindle, Weber, & Schmidt. p. 48.OCLC260534353. Also quoted byRicheson (2012),p. 17, cited to Eves.
^Morris, Imogen I. (24 October 2023).Mechanising Euler's use of Infinitesimals in the Proof of the Basel Problem (PhD thesis). University of Edinburgh.doi:10.7488/ERA/3835.
^Hopkins, Brian;Wilson, Robin (2007). "Euler's science of combinations".Leonhard Euler: Life, Work and Legacy. Stud. Hist. Philos. Math. Vol. 5. Amsterdam: Elsevier. pp. 395–408.MR3890500.
^Euler, Leonhard (1757)."Principes généraux de l'état d'équilibre d'un fluide" [General principles of the state of equilibrium of a fluid].Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires (in French).11:217–273. Retrieved2021-06-12. Translated into English asFrisch, Uriel (2008). "Translation of Leonhard Euler's: General Principles of the Motion of Fluids".arXiv:0802.2383 [nlin.CD].
^Smith, D.P. and N. Keyfitz, (2013) Mathematical Demography: Selected Papers, Monographs, DOI 10.1007/978-3-642-35858-6_1 Springer-Verlag Demographic Research - Euler, L. (1760). 11. A general investigation into the mortality and multiplication of the human species. A General Investigation into the Mortality and Multiplication of the Human Species, Theoretical Population Biology 1: 307-314. Translated by Nathan and Beatrice Keyfitz
^Newell, Colin. (1988) Methods and models in demography. Belhaven Press.
^Inaba, Hisashi (2017) Chapter 1 The Stable Population Model inAge-structured population dynamics in demography and epidemiology. Springer Singapore.
^Lotka, A. J. (1907). Relation between birth rates and death rates. Science, 26(653), 21-22.
^Sharpe, F. R., & Lotka, A. J. (1911). L. A problem in age-distribution. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(124), 435-438.
^Lotka, A. J. (1922). The stability of the normal age distribution. Proceedings of the National Academy of Sciences, 8(11), 339-345.
^Tegg, Thomas (1829)."Binary logarithms".London encyclopaedia; or, Universal dictionary of science, art, literature and practical mechanics: comprising a popular view of the present state of knowledge, Volume 4. pp. 142–143. Retrieved2021-06-13.
^Gollin, Edward (2009). "Combinatorial and transformational aspects of Euler'sSpeculum Musicum". In Klouche, T.; Noll, Th. (eds.).Mathematics and Computation in Music: First International Conference, MCM 2007 Berlin, Germany, May 18–20, 2007, Revised Selected Papers. Communications in Computer and Information Science. Vol. 37. Springer. pp. 406–411.doi:10.1007/978-3-642-04579-0_40.ISBN978-3-642-04578-3.
^Bailhache, Patrice (17 January 1997)."La Musique traduite en Mathématiques: Leonhard Euler".Communication au colloque du Centre François Viète, "Problèmes de traduction au XVIIIe siècle", Nantes (in French). Retrieved2015-06-12.
^"E"(PDF).Members of the American Academy of Arts & Sciences, 1780–2017.American Academy of Arts and Sciences. pp. 164–179. Retrieved2019-02-17. Entry for Euler is on p. 177.
^abcdFerraro, Giovanni (2007). "Euler's treatises on infinitesimal analysis:Introductio in analysin infinitorum, institutiones calculi differentialis, institutionum calculi integralis". In Baker, Roger (ed.).Euler Reconsidered: Tercentenary Essays(PDF). Heber City, UT: Kendrick Press. pp. 39–101.MR2384378. Archived fromthe original(PDF) on 2022-09-12.
^Reviews ofIntroduction to Analysis of the Infinite:
Aiton, E. J. "Introduction to analysis of the infinite. Book I. Transl. by John D. Blanton. (English)".zbMATH.Zbl0657.01013.
Ştefănescu, Doru. "Euler, Leonhard Introduction to analysis of the infinite. Book I. Translated from the Latin and with an introduction by John D. Blanton".Mathematical Reviews.MR1025504.
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