Leonhard Euler’s Contributions in Mathematics

Leonhard Euler

A Great mathematician of the 18th century, Leonhard Euler was born on 15th April 1707 in Basel, Switzerland. Euler developed his interest in mathematics through the teachings of his father. Not only a mathematician but Euler was also a great astronomer, geographer, logician, and physician. He spent a great part of his life in Russia at St Petersburg. He was appointed as a senior chair of mathematics at the St Petersburg Academy of Sciences. In the field of mathematics, he made several significant contributions as he founded graph theory and studies of topology, number theory, complex analysis and infinitesimal calculus. He also gave an idea on how to represent a mathematical function. Representation of π, imaginary number ‘i’, Greek ∑ for summation and a constant, the base of the natural logarithm, e, also known as Euler’s constant. His other works include projects on the topics of cartography, science education, magnetism, fire engines, machines, and shipbuilding. He also studied the three-body problem, elasticity, wave theory of lights, hydraulics, and music. In his book, ‘Theory of the Motions of Rigid Bodie’, he introduced analytical mechanics. In 1735, Euler faced some health issues, and he almost lost his life by fever. Euler in his autobiographical writings says that,

 In 1738, with overstrain due to his cartographic work and that by 1740 I had lost an eye and the other currently may be in the same danger.”

After an illness, Euler became totally blind. In 1771, his house was destroyed by fire, and Euler somehow saved his life and mathematical works. He continued his work on optics, algebra, and lunar motion even in complete blindness. He completed almost half of his total works despite the total blindness. On 18th September 1783, he uttered only ‘I am dying’ before he lost consciousness. He died due to a brain haemorrhage. After his death, St Petersburg Academy published his works for 50 more years. His major works are mostly in the field of mathematics. His influence on mathematics is evident from the remarks made by his fellow mathematicians. Pierre Simon Laplace said that,

Read Euler, read Euler, he is the master of us all.”

And Carl Friedrich Gauss remarked that,

The study of Euler’s works will remain the best school for the different fields of mathematics, and nothing else can replace it.”

1. Mathematical Notation

Euler is credited for giving several mathematical notations used to date. He introduced notation f(x) to define a function. He gave the letter ‘e’ to represent the base of the logarithm. To denote the imaginary number √-1, he gave the notation ‘i’, and ‘π’ to denote the ratio of the circumference of a circle to its diameter. He also gave ‘∑’ for summation, modern notation ‘sin’, ‘cos’, ‘tan’ for trigonometric functions, and the notation for finite differences Δ, Δ² and many others.

2. Graph Theory and Topology

He laid the foundation of graph theory and topology while solving the problem of seven bridges of Königsberg. In 1736, he solved this problem. This problem is about the city of Königsberg in Prussia, set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem was to find a route through the city that would cross each bridge only once. Euler’s solution to this problem was considered the first theorem of graph theory. While finding out the solution, he made an important remark that the only useful information in the problem is the number of bridges and the list of their endpoints, which laid the foundation of topology.

3. Euler Characteristic

Euler characteristic is a number in algebraic topology that describe the shape and structure of topological space, irrespective of the way it is bent. Leonhard Euler discovered this concept for a convex polyhedron, and used it to prove many theorems about polyhedra. It is denoted by the Greek letter χ. The formula of Euler characteristic for convex polyhedra is known as Euler formula and is given by,

χ = V-E+F, where V denotes vertices, E denotes edges, and F denotes faces of polyhedra. For any convex polyhedron, the Euler characteristic is 2. It may variate and can be negative for non-convex polyhedrons. For projective polyhedra, the Euler characteristic is 1 and for toroidal polyhedra, it is 0. Euler characteristic of plane graphs can be determined by the same Euler formula, and the Euler characteristic of a plane graph is 2.

4. Euler’s Path and Circuit

Euler’s trial or path is a finite graph that passes through every edge exactly once. Euler’s circuit of the cycle is a graph that starts and end on the same vertex. This path and circuit were used by Euler in 1736 to solve the problem of seven bridges. Euler, without any proof, stated a necessary condition for the Eulerian circuit. He said that for the existence of an Eulerian circuit, the graph should be connected with all the vertices of even degree. He further claimed, that for the existence of Eulerian trails, it is necessary that zero or two vertices have an odd degree. If there are no odd degree vertices, then Euler’s trials are circuits. A graph is called semi-Eulerian when it has an Eulerian trial but no Eulerian circuit.

Euler path

5. Introduction of Beta and Gamma Functions

Beta and Gamma functions were introduced and used by Leonhard Euler in applied mathematics. Gamma function is an extension to the factorial operation (n!), extending from positive real to even complex values of n. The relation is given by,

Γ(n) = (n-1)!

The beta function is also known as Euler’s integral of the first kind and is defined by,

BETA functions

for complex numbers m and n, such that Re m >0 and Re n>0. Also, beta function is symmetric, Β(x, y) = Β(y, x). Also, beta and gamma functions are closely related and the relation is given by, Β(x, y)= Γ(x)Γ(y)⁄Γ(x+y).

6. Euler’s method for Solving Differential Equations

Euler’s method, also known as the forward Euler method, is a first-order method for solving ordinary differential equations with some given initial value. It is the simplest form of the Runge-Kutta method. This method was mentioned by Euler in his book, “Institutionum calculi integralis”. Given the equation of type y´(t) = f(t, y(t)), y(t0)=y0. Let h be the step size and set tn= t0+nh, then the Euler method is given by, yn+1 =  yn + hf(tn+yn).

7. Solution to Basel’s Problem

Basel’s problem was considered to be one of the most difficult problems in mathematics. This problem was about the sum of an infinite series that is the summation of the reciprocals of the squares of the natural numbers. Several mathematicians tried to solve this problem, and Euler succeeded by solving it in 1734. Euler provided the exact solution of Basel’s problem as,

basel problem

The above-calculated sum is approximated to 1.644936. The solution of this problem is used to calculate the probability of two large random numbers being prime. In the range of 1 to n, as n tends to infinity, two random numbers are relatively prime with the probability of 6⁄π², reciprocal of Basel’s problem.

8. Number Theory

There are several results in number theory that are proven by Leonhard Euler.

Fermat’s conjecture

Fermat’s conjecture states that the number {2}^{n} + 1 are always prime if n is a power of 2. In 1729, Goldbach asked Euler about this conjecture, and after that Euler started verifying it for n= 1, 2, 4, 8, and 16. In 1732, he showed that {2}^{32}+1 =  is divisible by 641 and so is not prime. He proved one more result given by Fermat that states,

if a and b are coprime then a² + b²  has no divisor of the form

Φ Function

He introduced the Euler phi function Φ(n) that will give the number of integers k such that 1≤k≤n and k are coprime to n.

Zeta Function and Prime Numbers

In 1737, he gave the very famous relation of the zeta function and prime numbers. The relation is given by,

ζ(s) = ∑ 1⁄{n}^{s} = 1⁄1{p}^{-s}, where n denotes the natural numbers and p denotes the prime numbers.

Fermat’s Last theorem​

Euler gave the proof of Fermat’s last theorem for n=3. The most significant fact about the proof was that his proof involved numbers of form a+b√-3 for integers a and b.

Quadratic Reciprocity

Conjecture of the law of quadratic reciprocity was led by Euler and proved by Gauss. This law states modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. He also worked on perfect numbers and prime number theorem.

Euler’s partition theorem

Euler’s partition theorem states,

The number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1.”

9. Analysis

Euler and Bernoulli had made significant contributions in analysis and were responsible for the fast progress in this field of mathematics.

Complex Analysis

Euler found the way of solving integrals having complex limits, and thus led to the foundation of complex analysis. He further states that for any real number φ, the complex exponential function satisfies the equation,

{e}^{iφ} = cos φ + isin φ and Euler’s identity is a special case of this given by,

{e}^{iπ} + 1 = 0, and is called as the most remarkable formula in mathematics by Rechard Feynman.

Logarithmic function

Euler devised various ways in which logarithmic functions can be used in applied mathematics. He defined logarithms for complex and negative numbers and expressed logarithmic functions in the form of power series. Initially, several mathematicians believed that log(x)= log(-x) for any real x. In 1747, Euler proved the natural logarithm of -1 as iπ.

Power Series of ‘e’ and Inverse Tangent Function

Euler is credited for introducing the power series expansion of ‘e’ and inverse tangent function:

Power series of e

Power series of inverse tangent function

Analytical Number Theory

In this new field of mathematics, Euler introduced the hypergeometric series, q-series, hyperbolic trigonometric functions, and analytical theory of continued fractions. Using divergence of harmonic series, Euler proved the infinitude of primes, and his works in this field led to the introduction of the prime number theorem.

10. Calculus of Variation

In 1766, Euler coined the term calculus of variations. This field of mathematics analysis deals with the small changes in functions and functionals to find maxima and minima of functionals. Euler, along with an Italian mathematician, Joseph Louis Lagrange, developed the most famous result known as Euler- Lagrange equation.

11. Euler’s Theorem

Euler’s theorem is also known as the Euler-Fermat theorem or Euler’s totient theorem. This theorem is an extension of Fermat’s little theorem, which has a restriction of ‘n’ being prime. In 1763, Euler proved the generalisation of the theorem, where n is not necessarily prime. This theorem states,

 If n and a are coprime positive integers, and φ(n) is Euler’s totient function, then a raised to the power φ(n) is congruent to 1 modulo n; that is

{a}^{φ(n)} ≡ 1 modulo n

12. Differential Geometry

Problems in the mathematical analysis led Euler to make significant contributions in differential geometry. He worked on linear equations with constant coefficients, second-order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. He also considered the theory of surfaces and the curvature of surfaces. Many unpublished results by Euler in this area were rediscovered by Gauss. He also introduced the Mascheroni constant (γ) to facilitate the use of the differential equations and the constant γ is given by,

Mascheroni constant

13. Applied Mathematics

To make the numerical approximation of integrals easy, Euler proposed various approximations, and one of them is Euler- Maclaurin formula which is now known as Darboux’s formula. This formula calculates the difference between an integral and a closely related sum. Euler used this formula to compute those infinite series that converge slowly, whereas Maclaurin used it to calculate integrals. The formula states,

If m and n are natural numbers and f(x) is a real or complex-valued continuous function for real numbers x in the interval [m,n], then the integral

Euler integral

can be approximated by the sum as S = f(m+1)+…………+f(n-1)+f(n).

14. Euler’s Four Square Identity

Euler’s four-square identity states,

The product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.”

This identity was used by Lagrange to prove his four-square theorem. Algebraically, this identity can be written as,

Euler four square identity

15. Theorems by Euler

  • Euler’s homogeneous function theorem – Suppose that the function f : {R}^{n} /{0} → R is continuously differentiable. Then f is positively homogeneous of degree k if and only if, x.∇f(x) = kf(x).
  • Euler’s rotation theorem – In 1775, with the help of spherical geometry, Euler stated that in a three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. In simple words, it means that the composition of two rotations is also a rotation.
  • Euler’s theorem of differential geometry – This theorem proves the existence of principal curvature and gives us the direction in which the surface curves the most and the least. Geometrically, for a three-dimensional Euclidean space M, there is a point p on M. A normal plane through p, is a plane passing through p having a normal vector to M. A normal plane {P}_{x}, passes through each tangent vector to M at p, cuts a curve in M. Suppose curvature of the curve be {k}_{X}. Considering that all {k}_{x} are not equal, there is some unit vector X1 for which k1 = κX1 is as large as possible and another unit vector X2 for which k2 = κX2 is as small as possible. According to Euler’s theorem, {X}_{1} and {X}_{2} are perpendiculars, and if X is any vector making an angle θ with X1, then {k}_{X1} = k1cos²θ + k2sin²θ, where k1 and k2 are principal curvatures and X1 and X2 are the corresponding principal directions.
  • Euler’s theorem in geometry – Euler’s theorem states that the distance d between the circumcenter and incenter of a triangle is given by d² = R(R-2r), where R and r denote the circumradius and inradius respectively.
  • Euler’s quadrilateral theorem – Euler’s law on quadrilaterals establishes a relation between the convex quadrilateral and its diagonals. The theorem states, For a convex quadrilateral with sides a, b, c, d, diagonals e, f and g being the line segment connecting the midpoints of the two diagonals then the equation a² + b² + c² + d² = e² + f² + 4g² holds.
  • Euclid- Euler theorem – This theorem states that an even number is perfect if and only if it has the form 2p−1(2p − 1), where 2p − 1 is a prime number.



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