Euler & Laplace
The most prolific mathematician in history and the French Newton who tamed probability and celestial mechanics
14.1 Leonhard Euler (1707 – 1783)
Leonhard Euler was born on April 15, 1707 in Basel, Switzerland, the son of a Calvinist pastor who had studied mathematics under Jakob Bernoulli. The young Euler entered the University of Basel at the age of thirteen, where he came under the tutelage of Johann Bernoulli — arguably the finest mathematician in Europe at the time. Bernoulli quickly recognized Euler's extraordinary talent and began giving him private Saturday afternoon lessons, an honour the great professor extended to almost no other student.
Euler's Life — Key Dates
- 1707 — Born in Basel, Switzerland
- 1720 — Enters University of Basel aged 13
- 1727 — Joins the St. Petersburg Academy of Sciences
- 1735 — Loses sight in his right eye (possibly from overwork or an infection)
- 1741–1766 — Works at the Berlin Academy under Frederick the Great
- 1766 — Returns to St. Petersburg; loses sight in his left eye shortly after
- 1771 — Completely blind; dictates an average of one paper per week
- 1783 — Dies in St. Petersburg; “He ceased to calculate and to live”
At twenty, Euler moved to the newly founded St. Petersburg Academy of Sciences in Russia, where he would spend two enormously productive decades. He mastered every branch of mathematics then known — number theory, algebra, analysis, geometry, mechanics, optics, and astronomy — and created several new ones. In 1741, Frederick the Great lured him to the Berlin Academy, where he remained for twenty-five years before returning to St. Petersburg at the invitation of Catherine the Great.
The sheer volume of Euler's output is almost incomprehensible. His collected works, the Opera Omnia, fill over 80 quarto volumes containing roughly 866 distinct publications. He is credited with approximately one-third of all the mathematics, physics, and engineering published in the 18th century. Even after going completely blind in 1771, Euler continued to produce mathematics at an astonishing rate, dictating to his assistants and computing vast expressions entirely in his head. It is said that on the day of his death — September 18, 1783 — he calculated the orbit of the recently discovered planet Uranus, drank tea, played with his grandchild, and then, as the mathematician Condorcet wrote, “he ceased to calculate and to live.”
Pierre-Simon Laplace famously advised students: “Read Euler, read Euler. He is the master of us all.” This was not empty praise — virtually every major branch of mathematics bears Euler's fingerprints. He was the rare genius who combined breathtaking computational skill with deep conceptual insight.
14.2 Euler's Notation — The Modern Language of Mathematics
More than any other individual, Euler is responsible for the mathematical notation we use today. Before Euler, mathematical writing was a patchwork of inconsistent symbols and verbal descriptions. Euler's genius for clear, elegant notation transformed mathematics into a universal language.
Notations Introduced or Popularized by Euler
- $f(x)$ — the modern function notation, introduced in 1734
- $e$ — the base of the natural logarithm, $e \approx 2.71828\ldots$
- $i$ — the imaginary unit, $i = \sqrt{-1}$
- $\pi$ — while not the first to use it, Euler's adoption made it standard
- $\Sigma$ — summation notation, $\sum_{k=1}^{n} a_k$
- $\sin, \cos, \tan$ — trigonometric abbreviations as functions
- $\Delta$ for finite differences, $\log$ for logarithms
Euler's textbooks — the Introductio in Analysin Infinitorum (1748), Institutiones Calculi Differentialis (1755), and Institutiones Calculi Integralis (1768–1770) — became the standard references for a century. The Introductio alone established the function concept as the central object of analysis, replacing the earlier focus on curves. In this work, Euler treated$e^x$, $\sin x$, and $\cos x$ as infinite series and derived their fundamental properties purely algebraically.
Before Euler, the exponential function was not clearly distinguished from the logarithm, and the relationship between exponentials and trigonometric functions was murky. Euler unified all of these through his masterful use of infinite series and complex numbers, revealing connections that previous mathematicians had scarcely imagined.
14.3 Euler's Identity — The Most Beautiful Equation
Euler's identity is widely regarded as the most beautiful equation in all of mathematics:
This single equation connects the five most fundamental constants in mathematics: $e$ (the base of natural logarithms), $i$ (the imaginary unit), $\pi$ (the ratio of a circle's circumference to its diameter), $1$ (the multiplicative identity), and $0$ (the additive identity). It also involves the three basic arithmetic operations: addition, multiplication, and exponentiation.
The identity is a special case of Euler's formula, which Euler published in 1748:
Setting $x = \pi$, we get $e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0 = -1$, so $e^{i\pi} + 1 = 0$.
Proof via Taylor Series
We derive Euler's formula from the Taylor series of the exponential and trigonometric functions. Recall the three fundamental series:
Now substitute $ix$ into the exponential series. Since $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, etc.:
Separating real and imaginary parts:
Euler's formula has far-reaching consequences. It provides the bridge between exponential and trigonometric functions, enables the polar form of complex numbers $z = re^{i\theta}$, simplifies the analysis of differential equations, and underpins signal processing, quantum mechanics, and electrical engineering. The formula also yields the beautiful corollaries:
These express trigonometric functions in terms of complex exponentials, a representation that is fundamental in Fourier analysis and modern physics.
14.4 The Basel Problem
The Basel problem, named after Euler's hometown, asked for the exact value of the sum of the reciprocals of the perfect squares:
This problem had been posed by Pietro Mengoli in 1644 and had defeated the greatest mathematicians of Europe for nearly a century, including the Bernoulli brothers. Jakob Bernoulli had computed the sum to six decimal places ($\approx 1.644934$) but could not find a closed form. In 1735, the 28-year-old Euler stunned the mathematical world by proving:
Euler's Proof Sketch: The Product Formula for Sine
Euler's brilliant and audacious argument treats $\sin x$ as an “infinite polynomial” with roots at $x = 0, \pm\pi, \pm 2\pi, \pm 3\pi, \ldots$ By analogy with finite polynomials, a polynomial with roots $r_1, r_2, \ldots$ can be written as $c(x - r_1)(x - r_2)\cdots$.
Since $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$, we can factor out $x$:
The function $\frac{\sin x}{x}$ has roots at $x = \pm n\pi$ for $n = 1, 2, 3, \ldots$, so Euler wrote:
Now expand the right side and compare the coefficient of $x^2$. On the left side, the coefficient of$x^2$ is $-\frac{1}{3!} = -\frac{1}{6}$.
On the right side, the coefficient of $x^2$ comes from picking the $-\frac{x^2}{n^2\pi^2}$ term from exactly one factor and $1$ from all others:
Equating coefficients:
While Euler's original argument was not fully rigorous by modern standards (treating $\sin x$as an infinite polynomial required the Weierstrass product theorem, developed a century later), the result was correct and profoundly important. It was the first time anyone had found an exact, closed-form value for $\zeta(2)$, linking an arithmetic series to the transcendental constant $\pi$.
Euler went on to evaluate $\zeta(2k)$ for all positive integers $k$, finding:
In general, $\zeta(2k) = \frac{(-1)^{k+1}(2\pi)^{2k}B_{2k}}{2(2k)!}$ where $B_{2k}$ are the Bernoulli numbers. The odd values $\zeta(3), \zeta(5), \ldots$ remain mysterious to this day — even $\zeta(3)$ (Apéry's constant) was only proved irrational in 1978.
14.5 Graph Theory — The Königsberg Bridges
In 1736, Euler solved a famous recreational puzzle and, in doing so, founded an entirely new branch of mathematics: graph theory. The city of Königsberg (now Kaliningrad) was built on both sides of the river Pregel, with two islands in the river. Seven bridges connected the various land masses. The question was: is it possible to walk through the city, crossing each bridge exactly once?
Euler abstracted the problem by replacing the land masses with vertices (nodes) and the bridges with edges. He then proved a general theorem:
Euler's Theorem on Traversability
A connected graph has an Eulerian circuit (a closed walk that traverses every edge exactly once) if and only if every vertex has even degree.
A connected graph has an Eulerian path (an open walk traversing every edge exactly once) if and only if it has exactly zero or two vertices of odd degree.
In the Königsberg graph, all four vertices have odd degree (3, 3, 3, and 5), so neither an Eulerian circuit nor an Eulerian path exists.
This was revolutionary for two reasons. First, it solved the problem definitively — no amount of clever route-finding would succeed. Second, it showed that the geometric layout of the bridges was irrelevant; only the combinatorial structure (the graph) mattered. This insight — that properties of a network depend only on its connectivity — is the foundational idea of topology and combinatorics.
Euler also discovered the remarkable polyhedron formula, relating the number of vertices $V$, edges $E$, and faces $F$ of any convex polyhedron:
Verifying the Polyhedron Formula
- Cube: $V = 8, \; E = 12, \; F = 6$ → $8 - 12 + 6 = 2$ ✔
- Tetrahedron: $V = 4, \; E = 6, \; F = 4$ → $4 - 6 + 4 = 2$ ✔
- Octahedron: $V = 6, \; E = 12, \; F = 8$ → $6 - 12 + 8 = 2$ ✔
- Dodecahedron: $V = 20, \; E = 30, \; F = 12$ → $20 - 30 + 12 = 2$ ✔
- Icosahedron: $V = 12, \; E = 30, \; F = 20$ → $12 - 30 + 20 = 2$ ✔
This formula, now written as $\chi = V - E + F = 2$ (where $\chi$ is the Euler characteristic), extends far beyond convex polyhedra. It is a topological invariant: for a torus (donut shape), $\chi = 0$; for a surface of genus $g$ (with $g$ holes),$\chi = 2 - 2g$. This generalization became a cornerstone of algebraic topology.
14.6 Number Theory
Euler made transformative contributions to number theory, a subject he loved deeply. He proved many results that Fermat had merely conjectured, and introduced concepts that remain central to the field.
Euler's Totient Function
For any positive integer $n$, the Euler totient function $\varphi(n)$ counts the number of integers from $1$ to $n$ that are relatively prime to $n$. For example:
- $\varphi(1) = 1$
- $\varphi(6) = 2$ (only 1 and 5 are coprime to 6)
- $\varphi(p) = p - 1$ for any prime $p$
- $\varphi(p^k) = p^k - p^{k-1} = p^{k-1}(p-1)$
For $n = p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, the totient is:
Euler's Theorem
If $\gcd(a, n) = 1$, then:
This generalizes Fermat's Little Theorem (the case $n = p$ prime, where $\varphi(p) = p - 1$):
Euler's theorem is the foundation of the RSA cryptosystem, one of the most widely used encryption algorithms in the modern world.
Example: Computing with Euler's Theorem
Find $7^{222} \pmod{10}$ (i.e., the last digit of $7^{222}$).
We have $\varphi(10) = 10(1 - \frac{1}{2})(1 - \frac{1}{5}) = 4$ and $\gcd(7, 10) = 1$.
By Euler's theorem: $7^4 \equiv 1 \pmod{10}$.
Since $222 = 4 \times 55 + 2$, we get $7^{222} = (7^4)^{55} \cdot 7^2 \equiv 1^{55} \cdot 49 \equiv 9 \pmod{10}$.
The last digit of $7^{222}$ is 9.
Among Euler's other number-theoretic achievements: he proved Fermat's theorem that every prime of the form $4k + 1$ can be written as a sum of two squares; he discovered the law of quadratic reciprocity (though he could not prove it — that honour fell to Gauss); and he made deep investigations into the partition function $p(n)$, which counts the number of ways to write $n$ as a sum of positive integers. Euler discovered the generating function:
and the remarkable pentagonal number theorem:
14.7 Analysis — The Gamma Function and Beyond
Euler's contributions to analysis are vast. Among the most important is his extension of the factorial function to non-integer arguments.
The Gamma Function
For $\text{Re}(s) > 0$, Euler defined:
This satisfies the recurrence $\Gamma(s+1) = s\,\Gamma(s)$, and since $\Gamma(1) = 1$, we get $\Gamma(n+1) = n!$ for all non-negative integers $n$. Thus the gamma function extends the factorial to the complex plane.
Special Values of the Gamma Function
- $\Gamma(1) = 0! = 1$
- $\Gamma(2) = 1! = 1$
- $\Gamma(1/2) = \sqrt{\pi}$ — a remarkable fact connecting factorials and $\pi$
- $\Gamma(3/2) = \frac{1}{2}\Gamma(1/2) = \frac{\sqrt{\pi}}{2}$
- $\Gamma(n + 1/2) = \frac{(2n)!}{4^n n!}\sqrt{\pi}$
Euler also introduced the beta function:
The Euler–Maclaurin summation formula provides a powerful bridge between discrete sums and continuous integrals:
where $B_{2j}$ are the Bernoulli numbers and $R_p$ is a remainder term. This formula is invaluable in numerical analysis, asymptotic expansions, and analytic number theory.
In differential equations, Euler developed the method of integrating factors, the theory of linear differential equations with constant coefficients (the “characteristic equation” method), and the Euler–Lagrange equation of the calculus of variations:
which finds functions that extremize integral functionals $\int F(x, y, y')\,dx$. This equation is the mathematical heart of classical mechanics, optics, and the principle of least action.
14.8 Pierre-Simon Laplace (1749 – 1827)
Pierre-Simon, marquis de Laplace, was born in Normandy, France, to a family of modest means. He studied at the University of Caen before traveling to Paris, where he impressed d'Alembert so thoroughly that he was immediately appointed professor of mathematics at the École Militaire. Laplace would go on to become the most influential applied mathematician of his era, earning the nickname the “French Newton.”
Laplace's Life — Key Dates
- 1749 — Born in Beaumont-en-Auge, Normandy
- 1771 — Elected to the French Academy of Sciences
- 1799–1825 — Publishes the five volumes of Mécanique Céleste
- 1812 — Publishes Théorie Analytique des Probabilités
- 1827 — Dies in Paris
Laplace's magnum opus, the Traité de Mécanique Céleste (five volumes, 1799–1825), systematized and extended Newton's gravitational theory to the entire solar system. Where Newton had shown that gravitation explained Kepler's laws, Laplace tackled the far more difficult problem of perturbations — the deviations caused by the gravitational pull of multiple planets on each other. He demonstrated the long-term stability of the solar system, showing that the observed irregularities in planetary orbits were periodic rather than cumulative.
When Napoleon asked Laplace why God did not appear in his great work, Laplace famously replied: “Sir, I had no need of that hypothesis.”
The Laplacian operator $\nabla^2$, which Laplace introduced in the context of gravitational potential theory, appears throughout modern physics:
Laplace's equation $\nabla^2 f = 0$ describes gravitational fields, electrostatic potentials, steady-state heat distributions, and fluid flows. Solutions to Laplace's equation are called harmonic functions.
14.9 The Laplace Transform
The Laplace transform converts a function of time into a function of a complex frequency variable, transforming differential equations into algebraic equations. It is one of the most powerful tools in engineering and applied mathematics.
The Laplace Transform
For a function $f(t)$ defined for $t \geq 0$, the Laplace transform is:
where $s$ is a complex number. The inverse transform recovers $f(t)$ from $F(s)$:
Common Laplace Transforms
$\mathcal{L}\{1\} = \frac{1}{s}$ for $s > 0$
$\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$ for $s > a$
$\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$ for $s > 0$
$\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}$
$\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$
The key property that makes the Laplace transform so powerful is its effect on derivatives:
Solving a Differential Equation with the Laplace Transform
Solve $y'' + 4y = 0$, with $y(0) = 1$, $y'(0) = 0$.
Taking the Laplace transform of both sides:
Taking the inverse Laplace transform:
In engineering, the Laplace transform leads to the concept of a transfer function $H(s) = \frac{Y(s)}{X(s)}$, which characterizes a linear system's input-output relationship entirely in the frequency domain. This is the foundation of control theory and signal processing.
14.10 Probability and Statistics
Laplace's Théorie Analytique des Probabilités (1812) was the first comprehensive mathematical treatment of probability theory. Building on the earlier work of Pascal, Fermat, and the Bernoullis, Laplace systematized the entire subject and added powerful new tools.
Laplace defined probability as follows: if an event can occur in $m$ equally likely ways out of a total of $n$ equally likely outcomes, then the probability of the event is $\frac{m}{n}$. While this “classical” definition has limitations (it assumes equally likely outcomes), Laplace developed it into a sophisticated mathematical theory.
The Central Limit Theorem (Laplace's Version)
If $X_1, X_2, \ldots, X_n$ are independent random variables with the same distribution, each having mean $\mu$ and variance $\sigma^2$, then the standardized sum
converges in distribution to the standard normal distribution $N(0, 1)$ as $n \to \infty$. That is, $P(Z_n \leq z) \to \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z}e^{-t^2/2}\,dt$.
Laplace was also an early proponent of Bayesian inference. He formulated what we now call Bayes' theorem in its modern form:
where $P(H \mid E)$ is the posterior probability of hypothesis $H$ given evidence $E$, $P(E \mid H)$ is the likelihood, $P(H)$ is the prior probability, and $P(E)$ is the marginal likelihood. Laplace applied this extensively to problems in astronomy, demographics, and law.
Laplace's Demon
In 1814, Laplace articulated the most famous statement of scientific determinism:
“An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed — if this intellect were also vast enough to submit these data to analysis — it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future, just like the past, would be present before its eyes.”
This hypothetical all-knowing entity is known as Laplace's demon. Quantum mechanics and chaos theory have since shown that such complete determinism is impossible in practice — but the philosophical implications continue to be debated.
14.11 Legacy — Analysis as a Mature Discipline
Together, Euler and Laplace represent the culmination of 18th-century mathematics. Euler transformed analysis from a collection of techniques into a vast, unified discipline. He created the language and notation that every mathematician, scientist, and engineer uses today. His work in number theory, graph theory, mechanics, optics, and astronomy was equally groundbreaking. No mathematician before or since has contributed to so many different areas with such depth and originality.
Laplace brought mathematical analysis to bear on the physical world with unprecedented power. His celestial mechanics proved that the solar system was stable, his probability theory gave science a rigorous framework for reasoning under uncertainty, and his mathematical methods — the Laplace transform, spherical harmonics, potential theory — remain indispensable tools across all branches of science and engineering.
However, both men worked in an era when mathematical rigour was still developing. Euler freely manipulated divergent series and assigned them “values”; Laplace sometimes stated results without proof. The next generation — Cauchy, Abel, Weierstrass — would supply the rigorous foundations that Euler and Laplace's brilliant intuitions demanded. But the mathematical landscape they left behind was immeasurably richer than the one they had found.
As Gauss wrote of Euler: “The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it.” This remains as true today as it was two centuries ago.