Leonhard Euler

How to use this course

Leonhard Euler is the most prolific mathematician in history. The Eneström index lists 866 publications, and his collected works, the Opera Omnia, run to more than 80 volumes, with thousands of letters besides. Much of the notation of a first-year calculus course is his: \(e\), \(f(x)\), \(\sum\), \(i\), and the popular use of \(\pi\). Students meet his name in a dozen subjects, often without realizing it is the same man.

This course cannot cover everything. It follows seven threads: series and the exponential function, number theory, the first theorems of graph theory and topology, the calculus of variations, rigid-body mechanics, fluid dynamics, and celestial mechanics with optics. Each is chosen because a working mathematician or physicist still uses the result in the form Euler gave it.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Where Euler’s own argument was not rigorous by modern standards, the lesson says so and shows how the gap was later closed.

Timeline

Basel and Riehen, 1707–1727

Leonhard Euler was born in Basel on 15 April 1707. His father, Paul Euler, was a Protestant minister who had studied theology at Basel and had attended the mathematics lectures of Jakob Bernoulli. His mother, Marguerite Brucker, was also a pastor’s daughter. When Leonhard was about a year old the family moved to Riehen, a village just outside the city, where his father held the parish. Paul Euler taught his son elementary mathematics and expected him to enter the church.

Euler entered the University of Basel in 1720, still a boy. In 1723 he took a master’s degree in philosophy with a dissertation comparing the systems of René Descartes and Isaac Newton. His formal course was theology, Greek and Hebrew, but his interest lay elsewhere. Johann Bernoulli, then the leading mathematician in Europe, declined to give him private lessons. Instead he recommended books and let Euler visit once a week to ask about the passages he could not follow. The arrangement forced him to work alone and ask only when truly stuck.

Bernoulli eventually persuaded Paul Euler that his son was meant for mathematics, not the pulpit. Euler finished his studies in 1726. That year he wrote a short dissertation on the physics of sound, Dissertatio physica de sono, in support of an application for the vacant chair of physics at Basel. He was not appointed. In 1727 he entered the Paris Academy’s prize competition on the best way to place masts on a ship, and won second place behind Pierre Bouguer, a respectable result for a twenty-year-old who had never seen an ocean-going ship.

The real opening came from Russia. Peter the Great’s new Academy of Sciences in St Petersburg had recruited two of Johann Bernoulli’s sons, Nicolaus II and Daniel. When Nicolaus died in 1726, Daniel Bernoulli worked to bring Euler to the academy. The only post available was in physiology, and Euler spent his last months in Basel studying medicine. He left Basel on 5 April 1727.

St Petersburg, 1727–1734

Euler reached St Petersburg on 17 May 1727. The academy’s patron, Empress Catherine I, had just died, and the young institution faced years of political uncertainty. Euler was attached to the mathematical section rather than physiology, but he also served from 1727 to 1730 as a medical lieutenant in the Russian navy.

The academy was a good place for a young scientist. It paid its members to do research and publish it in the academy’s journal, the Commentarii. Euler shared lodgings with Daniel Bernoulli, and the two worked closely on mechanics, music theory and fluids. He became friends with Christian Goldbach, the academy’s secretary, who had a lively amateur interest in number theory. Their correspondence lasted until Goldbach’s death in 1764 and contains, among much else, the conjecture that every even number greater than 2 is a sum of two primes.

Goldbach turned Euler toward the problems of Pierre de Fermat. In 1732 Euler showed that Fermat had been wrong to believe that every number \(2^{2^n}+1\) is prime: the case \(n=5\) has the factor 641 (Lesson 05). In 1729 and 1730, in letters to Goldbach, he found the integral that extends the factorial to non-integer arguments, now the gamma function.

Euler became professor of physics around 1730. In 1733 Daniel Bernoulli returned to Basel, and Euler succeeded him in the senior chair of mathematics. On 7 January 1734 he married Katharina Gsell, daughter of a Swiss painter who taught at the academy’s art school. They had thirteen children, of whom only five survived infancy. Their eldest son, Johann Albrecht Euler, became a mathematician and astronomer.

The Basel problem years, 1734–1741

The next seven years made Euler famous. In 1735 he summed the reciprocals of the squares, a problem posed by Pietro Mengoli in 1650 that had defeated Jakob Bernoulli and the other leading analysts of the day. His answer, \(\pi^2/6\), was read to the academy on 5 December 1735, and he gave an argument accepted as a proof in 1741. On 26 August 1735 he presented the solution of the Königsberg bridge problem, the first theorem of graph theory. In 1737 he connected the primes to the zeta function through the product formula. All three appear in Part II.

In 1736 the academy published Mechanica, two volumes that recast Newton’s dynamics of a point mass in the language of differential equations. Where Newton argued with geometry, Euler wrote equations and solved them. It is also the first published work in which the letter \(e\) appears for the base of natural logarithms. The book set the programme for the rest of his life: to treat every branch of natural science by analysis.

Euler was also put to practical work. From 1735 he helped direct the academy’s geography department, preparing maps of the Russian Empire. In 1735 he suffered a severe fever and nearly died. By his own account, the overstrain of the cartographic work damaged his eyesight, and by 1738 he had lost most of the sight of his right eye. Portraits from later life show the damaged eye clearly.

He won a share of the Paris Academy’s prize in 1738 and again in 1740. Over his life he entered the competition fifteen times and won twelve. Yet conditions in Russia were worsening. After the death of Empress Anna in 1740 the regency was unstable and foreigners were viewed with suspicion. When Frederick II of Prussia, newly crowned, invited Euler to help refound the Berlin Academy, he accepted. He left St Petersburg on 19 June 1741 and arrived in Berlin on 25 July.

Analysis: the Basel problem and the exponential

For the general reader. Add \(1+\tfrac14+\tfrac19+\tfrac1{16}+\cdots\) and you approach a number near 1.645. For ninety years nobody knew what it was. Euler showed it is \(\pi^2/6\), so a sum of whole-number fractions secretly knows about circles. The same boldness with infinite expressions led him to the formula that joins the exponential function to trigonometry.

The product for the sine

A polynomial with roots \(r_1,\dots,r_m\) and constant term 1 factors as \(\prod(1-x/r_k)\). Euler applied this to \(\sin x/x\), whose roots are \(\pm\pi,\pm2\pi,\dots\), and wrote

Expanding the left side gives \(1-\frac{x^2}{6}+\frac{x^4}{120}-\cdots\). On the right, the coefficient of \(x^2\) is \(-\sum_n \frac{1}{n^2\pi^2}\). Equating the two:

The step from polynomials to an infinite product is not automatic: \(e^x\sin x/x\) has the same zeros. Euler knew the objection and found independent confirmations, including numerical ones to many decimal places. The product formula was proved rigorously in the nineteenth century, and Karl Weierstrass’s factorization theorem explains exactly when such products are valid.

The values \(\zeta(3),\zeta(5),\dots\) have no such formula. Euler tried and failed to find one; it is still unknown whether \(\zeta(5)\) is irrational.

The exponential and the circle

In the Introductio in analysin infinitorum (1748) Euler made the function the central object of analysis and defined \(e^x\) as \(\lim_{n\to\infty}(1+x/n)^n=\sum x^k/k!\). Substituting \(x=i\theta\) and separating even and odd powers,

Roger Cotes had reached an equivalent logarithmic statement in 1714, but Euler’s form made it a working tool. At \(\theta=\pi\) it gives \(e^{i\pi}+1=0\).

Gamma and gamma

In letters to Goldbach in 1729–30 Euler found a continuous extension of \(n!\). In the modern form, for \(x>0\),

and he proved the reflection formula \(\Gamma(x)\Gamma(1-x)=\pi/\sin(\pi x)\). In a 1734 paper on harmonic progressions he showed that \(H_n-\ln n\), with \(H_n=1+\tfrac12+\cdots+\tfrac1n\), tends to a constant, now written \(\gamma\approx0.5772156649\) and called the Euler–Mascheroni constant. Whether \(\gamma\) is irrational is unknown.

Number theory

For the general reader. Pierre de Fermat left a list of striking claims about whole numbers with almost no proofs. Euler, prompted by Goldbach, spent decades proving them, and found that some were false. Along the way he discovered that the primes are woven into the infinite series of Lesson 04, the starting point of analytic number theory.

Fermat’s little theorem and the totient

Fermat stated in 1640 that \(a^{p-1}\equiv1\pmod p\) for a prime \(p\nmid a\). Euler published the first proof, dated 1736. In work presented in 1758 and published in 1763 he generalized it, introducing the number \(\varphi(n)\) of integers in \(\{1,\dots,n\}\) coprime to \(n\). Carl Friedrich Gauss later fixed the notation \(\varphi(n)\).

Proof. Let \(r_1,\dots,r_{\varphi(n)}\) be the residues coprime to \(n\). Multiplication by \(a\) permutes them, so \(\prod(ar_i)\equiv\prod r_i\), that is \(a^{\varphi(n)}\prod r_i\equiv\prod r_i\pmod n\). The product is invertible modulo \(n\), so it cancels. In modern language: the group \((\mathbb Z/n)^\times\) has order \(\varphi(n)\), and Lagrange’s theorem applies.

Fermat numbers

Fermat believed every \(F_n=2^{2^n}+1\) is prime. \(F_0,\dots,F_4=3,5,17,257,65537\) are. Euler found in 1732 that \(F_5=4294967297=641\times6700417\). His method was to restrict the possible factors.

Proof. From \(2^{2^n}\equiv-1\pmod p\) we get \(2^{2^{n+1}}\equiv1\), so the order of 2 modulo \(p\) divides \(2^{n+1}\) but not \(2^n\): it equals \(2^{n+1}\). By Fermat’s theorem the order divides \(p-1\). For \(F_5\) the candidates are \(64k+1\): 193, 257, 449, 577 fail, and \(641=10\cdot64+1\) succeeds.

The Euler product

Proof sketch. Each factor is a geometric series \(1+p^{-s}+p^{-2s}+\cdots\). Multiplying the factors for \(p\le N\) produces \(\sum n^{-s}\) over all \(n\) whose prime factors are at most \(N\), each exactly once by unique factorization. Let \(N\to\infty\). For the divergence, take logarithms: \(\log\zeta(s)=\sum_p\sum_{k\ge1}\frac{p^{-ks}}{k}=\sum_p p^{-s}+R(s)\), where \(R(s)\le\sum_p\frac{p^{-2s}}{1-p^{-s}}\le\sum_n\frac{2}{n^2}\) stays bounded. As \(s\to1^+\), \(\zeta(s)\ge\sum_{n\le M}n^{-s}\to H_M\) for every \(M\), so \(\log\zeta(s)\to\infty\), and therefore \(\sum_p p^{-1}=\infty\). Euler expressed the growth as \(\log\log\) of the bound, a statement made precise by Franz Mertens in 1874.

This identity is the seed of Bernhard Riemann’s 1859 paper, which extended \(\zeta\) to complex \(s\), and of Peter Gustav Lejeune Dirichlet’s theorem on primes in arithmetic progressions.

Sums of two squares and beyond

Euler announced a proof to Goldbach on 6 May 1747 and published it in two papers. He also discovered the pentagonal number theorem, \(\prod_{n\ge1}(1-x^n)=\sum_{k\in\mathbb Z}(-1)^k x^{k(3k-1)/2}\), and stated, without proof, a form of the law of quadratic reciprocity that Gauss proved in 1796.

Graphs and polyhedra

For the general reader. The city of Königsberg had seven bridges joining four pieces of land. Could you walk through the city crossing each bridge exactly once? Euler saw that distances and shapes do not matter, only which pieces of land each bridge connects. That insight, that some questions depend only on connections, is the start of graph theory and topology.

Königsberg, 1735

The paper Solutio problematis ad geometriam situs pertinentis, presented on 26 August 1735 and published in 1741, labels the four land masses and the seven bridges. Euler treated the problem as one of “geometry of position”, a term from Leibniz, in which only incidence matters. In modern terms we have a multigraph with four vertices of degrees 5, 3, 3, 3.

Proof. Necessity (Euler): each time a walk passes through a vertex it uses two edges, so every vertex other than the start and end has even degree. Sufficiency, which Euler stated but did not prove, was proved by Carl Hierholzer: start anywhere and walk until stuck; with all degrees even you are stuck only at the start. If edges remain, some vertex on the closed walk touches an unused edge; build a second closed walk there and splice it in. Repeat. With two odd vertices, add an edge between them, find a closed walk, and remove the edge.

The polyhedron formula

In Elementa doctrinae solidorum, written in 1750 and published in 1758, Euler classified polyhedra by counting their vertices (“solid angles”), edges and faces, and stated the relation that bears his name. His proof was incomplete. Adrien-Marie Legendre and Augustin-Louis Cauchy gave proofs, and the search for a correct statement, with counterexamples such as a polyhedron with a tunnel through it, became the subject of Imre Lakatos’s Proofs and Refutations.

Proof (von Staudt’s tree argument). Choose a spanning tree \(T\) of the graph; it has \(V-1\) edges. Now form the dual graph, one vertex per face, joining two faces across each edge not in \(T\). This dual subgraph is connected, since \(T\) has no cycle to separate faces, and has no cycle, since such a cycle would separate vertices of \(T\). So it is a tree on \(F\) vertices with \(F-1\) edges. Every edge lies in exactly one of the two trees, so \(E=(V-1)+(F-1)\).

On a closed orientable surface of genus \(g\) the same count gives \(V-E+F=2-2g\). This number, the Euler characteristic, was the first topological invariant. It reappears in Gauss–Bonnet, in Poincaré’s theorem on vector fields, and in algebraic topology as the alternating sum of Betti numbers.

The calculus of variations

For the general reader. Ordinary calculus finds the point where a function is smallest. The calculus of variations finds the whole curve that makes a quantity smallest: the fastest slide, the shortest path on a surface, the shape of a soap film. Euler turned a collection of clever tricks into a general method, and found that the paths of moving bodies obey the same kind of principle.

From isolated problems to a method

Johann Bernoulli’s brachistochrone problem of 1696 asked for the curve of fastest descent. The Bernoulli brothers and others solved it and related isoperimetric problems by special arguments. Euler’s Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, completed in 1743 and published in Lausanne and Geneva in 1744, replaced the curve by a polygon, varied one vertex at a time, and passed to the limit. The result was a single differential equation valid for a whole class of problems, illustrated by about a hundred examples. In 1755 the nineteen-year-old Joseph-Louis Lagrange wrote to Euler with a cleaner method, varying the whole curve at once. Euler adopted it, praised it, and in 1766 named the subject the calculus of variations.

Derivation. Let \(\eta\) vanish at \(a,b\). Then

after integrating by parts; the boundary term vanishes. Since \(\eta\) is arbitrary, the bracket vanishes (the fundamental lemma of the calculus of variations). For the first integral, differentiate: \(\frac{d}{dx}\big(F-y'F_{y'}\big)=F_x+y'\big(F_y-\frac{d}{dx}F_{y'}\big)=F_x\), which is zero when \(F\) has no explicit \(x\).

Least action

In an appendix to the Methodus, Euler showed that a body moving under central forces follows a path along which \(\int v\,ds\) is stationary, the first precise statement of a principle of least action for orbits. Pierre-Louis Moreau de Maupertuis announced a more metaphysical version in the same year. When Samuel König disputed Maupertuis’s priority in 1751, Euler took Maupertuis’s side, an episode that did him little credit. Lagrange’s Mécanique analytique (1788) later built all of mechanics on such principles, and the Euler–Lagrange equations of a Lagrangian \(L(q,\dot q,t)\) are the equations of motion in every course on classical mechanics.

Rigid bodies

For the general reader. Newton described how a single particle moves. A spinning top, a planet with an equatorial bulge or a tumbling satellite are rigid bodies, and they move in ways that surprise everyone the first time. Euler wrote Newton’s law in the form \(F=ma\) used today, then extended it to rotation, giving the equations that still guide spacecraft.

A new principle, 1750

In Découverte d’un nouveau principe de mécanique, written in 1750 and published in 1752, Euler wrote the law of motion as three equations in fixed rectangular coordinates, \(M\ddot x=F_x\), \(M\ddot y=F_y\), \(M\ddot z=F_z\), and applied it to every particle of a body. This is the form in which Newton’s second law is now taught. In Du mouvement de rotation des corps solides autour d’un axe variable (written 1751, published 1765) he showed that every rigid body has three mutually perpendicular principal axes, and in Theoria motus corporum solidorum seu rigidorum (1765) he gave a full treatment, including the angles now named after him.

Euler’s equations

Work in a frame fixed in the body with axes along the principal axes through the centre of mass, so that the inertia tensor is \(\mathrm{diag}(I_1,I_2,I_3)\) and \(\mathbf L=(I_1\omega_1,I_2\omega_2,I_3\omega_3)\). For any vector, the rate of change in an inertial frame equals the rate in the rotating frame plus \(\boldsymbol\omega\times(\cdot)\). Hence \(\mathbf M=\dot{\mathbf L}_{\text{body}}+\boldsymbol\omega\times\mathbf L\), and componentwise:

With \(\mathbf M=0\) the motion has two conserved quantities, the kinetic energy \(T=\tfrac12\sum I_k\omega_k^2\) and \(|\mathbf L|^2=\sum I_k^2\omega_k^2\). The angular velocity therefore moves along the intersection of two ellipsoids, a picture due to Louis Poinsot. To recover the orientation itself one needs Euler’s angles \((\phi,\theta,\psi)\), which write any rotation as \(R=R_z(\phi)R_x(\theta)R_z(\psi)\).

The equations of fluids

For the general reader. Water and air have no fixed shape, so how can one write laws for them? Euler’s answer was to describe, at every point of space and every instant, the velocity, pressure and density of the fluid there, and to apply Newton’s law to each tiny parcel. The result is a set of partial differential equations that still define the subject, and whose mathematical behaviour is not fully understood today.

Field description

Daniel Bernoulli’s Hydrodynamica (1738) and Jean le Rond d’Alembert’s work on fluid resistance (1752) contained essential ideas, but neither had general equations. Euler wrote them in a series of Berlin memoirs; the decisive one, Principes généraux du mouvement des fluides, was written in 1755 and published in 1757. He treated pressure as a scalar field acting on every internal surface, a concept he had developed in his work on hydrostatics, and described the flow by a velocity field \(\mathbf u(\mathbf x,t)\).

Derivation. Mass in a fixed region \(\Omega\) changes only by flux through the boundary: \(\frac{d}{dt}\int_\Omega\rho=-\oint_{\partial\Omega}\rho\,\mathbf u\cdot\mathbf n\). The divergence theorem and the arbitrariness of \(\Omega\) give continuity. For momentum, a parcel’s acceleration is the material derivative \(\frac{D\mathbf u}{Dt}=\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u\). The pressure force on the parcel is \(-\oint p\,\mathbf n=-\int\nabla p\). Newton’s law for the parcel is the second equation. Viscosity is absent: adding it gives the Navier–Stokes equations of the next century.

In the 1757 memoir Euler also observed that a velocity potential \(\mathbf u=\nabla\phi\) is a special case, giving simple vortex flows as counterexamples. Whether smooth solutions of the three-dimensional equations can develop singularities in finite time is one of the central questions of modern analysis, closely tied to the Navier–Stokes Millennium Problem.

Proof. Use the identity \((\mathbf u\cdot\nabla)\mathbf u=\nabla\big(\tfrac12|\mathbf u|^2\big)-\mathbf u\times(\nabla\times\mathbf u)\). With \(\partial_t\mathbf u=0\), Euler’s equation becomes \(\nabla\big(\tfrac12|\mathbf u|^2+p/\rho+\Phi\big)=\mathbf u\times\boldsymbol\omega\), with \(\boldsymbol\omega=\nabla\times\mathbf u\). The right side is perpendicular to \(\mathbf u\), so the derivative along a streamline vanishes; if \(\boldsymbol\omega=0\) the gradient vanishes.

Moon, planets and lenses

For the general reader. In the eighteenth century, finding longitude at sea was a matter of life and death, and one method needed tables predicting the Moon’s position to a fraction of a degree years in advance. The Moon is pulled by both the Earth and the Sun, a three-body problem with no simple solution. Euler developed the mathematics that made usable tables possible, and also worked on how to build telescopes free of coloured fringes.

Planets and the Moon

Euler’s Theoria motuum planetarum et cometarum appeared in 1744. He won the Paris prize of 1748 on the mutual perturbations of Jupiter and Saturn, and his work for the 1752 contest on the same problem contains an early method of varying the orbital elements, which Lagrange and Laplace later made systematic. In 1747 Alexis Clairaut, d’Alembert and Euler each found that first-order theory gave only about half the observed motion of the Moon’s apsides; Clairaut showed in 1749 that higher-order terms remove the discrepancy, and Newton’s law survived.

Euler’s first lunar theory, Theoria motus lunae, appeared in 1753. Tobias Mayer of Göttingen built on Euler’s equations, fitted them to observations, and produced tables accurate enough for navigation. When the British Parliament rewarded the work in 1765, Mayer’s widow received £3,000 and Euler £300. Euler’s second lunar theory of 1772, written when he was blind, used rotating coordinates and ran to 775 pages.

Variation of constants

The idea is to keep the form of the unperturbed solution and let its constants change slowly. For a linear equation \(y''+y=f(t)\), write \(y=c_1(t)\cos t+c_2(t)\sin t\) with the side condition \(c_1'\cos t+c_2'\sin t=0\). Substituting gives \(-c_1'\sin t+c_2'\cos t=f\), a linear system for \(c_1',c_2'\). In celestial mechanics the “constants” are the six elements of a Kepler ellipse.

The collinear solutions

In De motu rectilineo trium corporum se mutuo attrahentium (written 1763, published 1767) and related papers, Euler found motions in which three bodies stay on a rotating straight line, each orbiting the centre of mass with the same period. In the restricted problem these are the points now called \(L_1,L_2,L_3\).

The condition says that gravity from both primaries balances the centrifugal term in the rotating frame. Lagrange added the two equilateral solutions \(L_4,L_5\) in 1772.

Optics

Euler defended a wave theory of light against Newton’s corpuscular view in Nova theoria lucis et colorum (1746). In 1747 he argued that colour fringes in lenses could be cancelled by combining media of different dispersion, such as glass and water, against Newton’s statement that this was impossible. John Dollond objected in 1753, relying on Newton. After Samuel Klingenstierna showed that Newton’s dispersion law did not fit the facts, Dollond experimented, succeeded in 1757 with two kinds of glass, and patented the achromatic lens in 1758. Euler’s three-volume Dioptrica (from 1769) set out the theory of optical instruments.

This follows from the lensmaker’s formula \(\phi_i=(n_i-1)K_i\), where \(K_i\) depends only on the curvatures: the change in power between the two wavelengths is \(\delta\phi_i=(n_{F}-n_{C})K_i=\phi_i/V_i\).

Berlin, 1741–1766

Frederick II wanted the Berlin Academy to rival those of Paris and London. When it was reorganized in 1744, Euler became director of its mathematics class, serving under its president, Maupertuis. He was the academy’s workhorse. Besides about 380 articles in twenty-five years, he oversaw the observatory and the botanical garden, advised on state lotteries, insurance and pensions, and worked on the hydraulics of the fountains at Sans Souci. After Maupertuis died in 1759 he ran the academy, though Frederick never gave him the title of president.

The Berlin years produced the great textbooks: the Methodus inveniendi (1744), the Introductio in analysin infinitorum (1748) and the Institutiones calculi differentialis (1755), along with the memoirs on mechanics, fluids, the Moon and optics described in Part II. He kept his ties to Russia, continued to publish in the St Petersburg journal and trained Russian students sent to Berlin, among them Semyon Kotelnikov and Stepan Rumovsky. In 1755 the Paris Academy elected him a foreign member.

Between 1760 and 1762 he wrote 234 letters in French to Friederike Charlotte of Brandenburg-Schwedt, a niece of Frederick, and her sister, on light, sound, gravity, logic, astronomy and philosophy. Published from 1768 as Letters to a German Princess, they were translated into many languages and became his most widely read work.

His relations with the king were poor. Frederick valued wit and polished conversation and found the pious, plain-spoken Euler dull. Euler had argued with d’Alembert on scientific matters for years, and in 1763 Frederick offered d’Alembert the presidency of the academy; d’Alembert declined, but the offer and Frederick’s interference in the academy’s affairs made Euler’s position uncomfortable. Catherine II of Russia offered generous terms, and in 1766, over Frederick’s angry objections, Euler returned to St Petersburg.

Return, blindness, death and legacy, 1766–1783

Soon after his return Euler fell ill and became almost totally blind; a cataract had formed in his good left eye. In 1771 a fire destroyed his house, and he escaped with his manuscripts. A cataract operation that year restored his sight for a few days, then he lost it entirely. His wife Katharina died in 1773, and in 1776 he married her half-sister, Salome Abigail Gsell.

Blindness hardly slowed him. With a phenomenal memory and the ability to calculate in his head, he dictated to his son Johann Albrecht, to Anders Johan Lexell, Wolfgang Krafft and later Nicolas Fuss, who arrived from Basel in 1772 and married Euler’s granddaughter. Almost half of his total output dates from after 1766, including the Institutiones calculi integralis (1768–70), the Dioptrica, the second lunar theory of 1772 and the popular Vollständige Anleitung zur Algebra (Elements of Algebra). He shared the Paris prizes of 1770 with Johann Albrecht and of 1772 with Lagrange.

On 18 September 1783 he spent the morning teaching mathematics to a grandchild, worked out on his slates the ascent of balloons, recently flown in Paris, and discussed with Lexell and Fuss the orbit of the newly discovered planet Uranus. At about five in the afternoon he suffered a brain haemorrhage and died late that evening, aged 76. In his memorial address the Marquis de Condorcet wrote that Euler cessa de calculer et de vivre, ceased to calculate and to live. He was buried in the Smolensk Lutheran Cemetery; in 1957 his remains were moved to the Lazarevskoe Cemetery of the Alexander Nevsky Monastery. The St Petersburg Academy went on publishing his backlog of papers for some fifty years.

Legacy

• Analysis: the function concept, the exponential and trigonometric functions as analytic objects, special functions, divergent series used with insight, and the textbooks from which Lagrange, Laplace and Gauss learned.
• Number theory: the first proofs of many of Fermat’s claims, the totient, and the Euler product, from which Riemann and Dirichlet built analytic number theory.
• Topology and combinatorics: graph theory, the Euler characteristic, the pentagonal number theorem and partitions.
• Physics: the modern form of Newton’s second law, rigid-body dynamics, the field description of fluids, the variational principles that run through all of theoretical physics, lunar theory and optics.
• Publication: the Opera Omnia, begun in Switzerland in 1911, has issued more than 80 volumes; the Euler Archive makes his papers freely available online.

Connections

People linked to Leonhard Euler across this collection. A name without a link has no page here yet. See Leonhard Euler in the network of all 337 lives →

Teachers and advisors

• Johann Bernoulli 1720-1726
• Werner Heisenberg

Students

• Joseph-Louis Lagrange 1754-1783
• Johann Friedrich Hennert 1766
• Nicolas Fuss 1772-1783
• Semyon Kotelnikov
• Stepan Rumovsky

Influenced by

• Abraham de Moivre
• Diophantus of Alexandria
• Euclid
• Gottfried Wilhelm Leibniz
• Isaac Newton
• Jacob Bernoulli
• Pierre de Fermat

Influenced

• Adrien-Marie Legendre
• Bernhard Riemann
• Carl Friedrich Gauss
• Carl Gustav Jacob Jacobi
• Claude-Louis Navier
• Gaspard-Gustave Coriolis
• Julius Robert Mayer
• Niels Henrik Abel
• Pafnuty Chebyshev
• Pierre-Simon Laplace
• Sofia Kovalevskaya
• Sophie Germain

Colleagues and collaborators

• Alexis Clairaut
• Colin Maclaurin 1740
• Daniel Bernoulli 1727-1733
• Johann Heinrich Lambert 1760-1766
• Pierre-Louis Moreau de Maupertuis 1744-1759
• Émilie du Châtelet
• Anders Johan Lexell 1768-1783

Rivals

• Jean le Rond d'Alembert

Friends

• Daniel Bernoulli 1727-1782
• Christian Goldbach 1727-1764

Family

• Johann Albrecht Euler

Also mentioned in this life

• André Weil
• Augustin-Louis Cauchy
• René Descartes
• Peter Gustav Lejeune Dirichlet
• Johannes Kepler
• Henri Poincaré
• George Gabriel Stokes
• Evangelista Torricelli
• John Wallis
• Karl Weierstrass

Fields

• analysis 20 lives
• number theory 45 lives
• graph theory 4 lives
• calculus of variations 9 lives
• mechanics 27 lives
• fluid dynamics 6 lives
• celestial mechanics 16 lives
• optics 38 lives

On CoursesHub

• Euler & Laplace · History of Mathematics
• Euler & Analytical Mechanics · History of Mathematics & Physics
• Variational Calculus · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Leonhard Euler, MacTutor History of Mathematics.
2. Leonhard Euler, Wikipedia (with extensive references).
3. R. S. Calinger, Leonhard Euler: Mathematical Genius in the Enlightenment, Princeton University Press, 2016.
4. E. A. Fellmann, Leonhard Euler, Birkhäuser, 2007.
5. W. Dunham, Euler: The Master of Us All, Mathematical Association of America, 1999.
6. The Euler Archive: Euler’s works by Eneström number, with originals and translations.
7. Leonhard Euler, Mathematics Genealogy Project.
8. Euler: genius blind astronomer mathematician, arXiv:1406.7397 (Euler’s astronomy, with dates).

Mathematics and physics

1. L. Euler, De summis serierum reciprocarum (E41), 1735, publ. 1740.
2. L. Euler, Solutio problematis ad geometriam situs pertinentis (E53), 1735, publ. 1741.
3. L. Euler, Methodus inveniendi lineas curvas (E65), Lausanne and Geneva, 1744.
4. L. Euler, Découverte d’un nouveau principe de mécanique (E177), 1750, publ. 1752.
5. L. Euler, Principes généraux du mouvement des fluides (E226), 1755, publ. 1757.
6. L. Euler, Elementa doctrinae solidorum (E230), 1750, publ. 1758.
7. L. Euler, Du mouvement de rotation des corps solides autour d’un axe variable (E292), 1751, publ. 1765.
8. L. Euler, De motu rectilineo trium corporum se mutuo attrahentium (E327), 1763, publ. 1767.
9. W. Dunham, Euler: The Master of Us All, chapters on number theory, analysis and combinatorics.
10. A. Weil, Number Theory: An Approach Through History from Hammurapi to Legendre, Birkhäuser, 1984.
11. C. Truesdell, Essays in the History of Mechanics, Springer, 1968.
12. I. Lakatos, Proofs and Refutations, Cambridge University Press, 1976.
13. D. Eppstein, Twenty-one proofs of Euler’s formula.