Astounding. Breathtaking. Inspiring. Captivating. Exhilarating.
Simply put, vehemently versatile + feverishly fascinating.
If there’s one mathematician that every other mathematician aspires to be, it’s Leonhard Euler. If that name sparks the Euler’s identity e^{iπ} + 1 = 0 in your mind, consider yourself an artist as not many people have the romantic inclination to appreciate the most beautiful equation in the history of mathematics.
Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastor’s daughter. He had two younger sisters: Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich.^{}
Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler’s name is associated with a large number of topics.
Euler is the only mathematician to have two numbers named after him: the important Euler’s number in calculus, e, approximately equal to 2.71828, and the Euler–Mascheroni constant γ (gamma) sometimes referred to as just “Euler’s constant”, approximately equal to 0.57721. It is not known whether γ is rational or irrational.
Mathematical notation
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 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 did not originate with him.
Analysis
The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler’s work. While some of Euler’s proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as
Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):^{[33]}
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 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
A special case of the above formula is known as Euler’s identity,
called “the most remarkable formula in mathematics” by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, i and π. In 1988, readers of the Mathematical Intelligencer voted it “the Most Beautiful Mathematical Formula Ever”. In total, Euler was responsible for three of the top five formulae in that poll.
De Moivre’s formula is a direct consequence of Euler’s formula.
Number theory
Euler’s interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Euler’s early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermat’s ideas, and disproved some of his conjectures.
Euler linked 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 the prime numbers; this is known as the Euler product formula for the Riemann zeta function.
Euler proved Newton’s identities, Fermat’s little theorem, Fermat’s theorem on sums of two squares, and he made distinct contributions to Lagrange’s foursquare theorem. He also invented the totient function φ(n), the number of positive integers less than or equal to 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 perfect numbers and Mersenne primes earlier proved by Euclid was onetoone, 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 of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. By 1772 Euler had proved that 2^{31} − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.
Graph Theory
Map of Königsberg in Euler’s time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.
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 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, specifically of planar graph theory.
Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges and faces of a convex polyhedron,^{[40]} 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.^{[41]} The study and generalization of this formula, specifically by Cauchy^{[42]} and L’Huillier, is at the origin of topology.
Applied mathematics
Some of Euler’s greatest successes were in solving realworld problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Venn diagrams, 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 the Euler approximations. The most notable of these approximations are Euler’s method and the Euler–Maclaurin formula. He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant:
One of Euler’s more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical 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.
Physics and astronomy
Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.
In addition, Euler made important contributions in optics. He disagreed with Newton’s corpuscular theory of light in the Opticks, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.
In 1757 he published an important set of equations for inviscid flow, that are now known as the Euler equations. In differential form, the equations are:
where
 ρ is the fluid mass density,
 u is the fluid velocity vector, with components u, v, and w,
 E = ρ e + ½ ρ (u^{2} + v^{2} + w^{2}) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,
 p is the pressure,
 ⊗ denotes the tensor product, and
 0 being the zero vector.
Euler is also well known in structural engineering for his formula giving the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness:
where
 F = maximum or critical force (vertical load on column),
 E = modulus of elasticity,
 I = area moment of inertia,
 L = unsupported length of column,
 K = column effective length factor, whose value depends on the conditions of end support of the column, as follows.

 For both ends pinned (hinged, free to rotate), K = 1.0.
 For both ends fixed, K = 0.50.
 For one end fixed and the other end pinned, K = 0.699…
 For one end fixed and the other end free to move laterally, K = 2.0.
 K L is the effective length of the column.
Logic
Euler is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.^{}
An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or “zones”: the interior, which symbolically represents the elements 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 settheoretic relationships (intersection, subset and disjointness). Curves whose interior zones do not intersect represent disjoint 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 (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it. Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since then, they have also been adopted by other curriculum fields such as reading.^{}
Music
Even when dealing with music, Euler’s approach is mainly mathematical. 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 did not leave him throughout his life.
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 2^{m}A, where A is the “exponent” of the genre (i.e. the sum of the exponents of 3 and 5) and 2^{m} (where “m is an indefinite number, small or large, so long as the sounds are perceptible”), 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, 2^{m}.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2^{m}.5, major third + minor sixth (C–E–C); the fourth is 2^{m}.3^{2}, two fourths and a tone (C–F–Bb–C); the fifth is 2^{m}.3.5 (C–E–G–B–C); etc. Genres 12 (2^{m}.3^{3}.5), 13 (2^{m}.3^{2}.5^{2}) and 14 (2^{m}.3.5^{3}) are corrected versions of the diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2^{m}.3^{3}.5^{2}) is the “diatonicochromatic”, “used generally in all compositions”, and which turns out to be identical with the system described by Johann Mattheson. Euler later envisaged the possibility of describing genres including the prime number 7.
Euler devised a specific graph, the Speculum musicum, to illustrate the diatonicochromatic genre, and discussed paths in this graph for specific intervals, reminding his interest for the Seven Bridges of Königsberg. The device knew a renewed interest as the Tonnetz in neoRiemannian theory.
Euler further used the principle of the “exponent” to propose a derivation of the gradus suavitatis (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only. Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form
 ds = Σ (k_{i}p_{i} – k_{i}) + 1
where p_{i} are prime numbers and k_{i} their exponents.^{}
Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field.
He spent most of his career in St. Petersburg, Russia. He was one of the most prolific mathematicians of all time, according to the U.S. Naval Academy (USNA), with 886 papers and books published. Much of his output came during the last two decades of his life, when he was totally blind. There was so much work that the St. Petersburg Academy continued publishing his work posthumously for more than 30 years.
William Dunham, Professor of Mathematics at Muhlenberg College, has given this great talk on Leonhard Euler which summarizes his mathematical acumen succinctly. Enjoy and get inspired :).