Euler and Gauss — Two Giants of the 18th and 19th Centuries

Two Summits of the 18th–19th Centuries

Building on Newton and Leibniz’s calculus, the 18th and early 19th centuries see mathematics systematized by two giants:

• Leonhard Euler (1707–1783) — born in Switzerland; the most prolific mathematician in history. “He wrote half of all mathematics himself.”
• Carl Friedrich Gauss (1777–1855) — born in Germany; the “Prince of Mathematicians.” Ranked with Archimedes and Newton as one of the three greatest of all time.

Euler was the giant who wrote incessantly; Gauss was the giant who held back publication. The contrast in publishing habits mirrors their opposite temperaments.

Leonhard Euler — “The 18th-Century Newton”

Leonhard Euler (1707–1783) was born in Basel, Switzerland — mathematician, physicist, astronomer, engineer. The most prolific mathematician in history; dozens of theorems, formulas, and concepts carry his name. He is called “the foundation of all mathematics” and “the 18th-century Newton.”

Life

• 15 April 1707 — Born in Basel; father Paul Euler was a Protestant pastor
• 1720 — Enters Basel University at 13; theology, philosophy, mathematics. Studies under Johann Bernoulli (Leibniz’s disciple)
• 1726 — Doctorate at 19, on acoustics
• 1727 — Invited to the St. Petersburg Academy at 20
• 1735 — Loses sight in his right eye (overwork on cartographic assignments)
• 1741 — Amid Russian political instability, accepts Frederick the Great’s invitation to the Berlin Academy
• 1766 — Returns to St. Petersburg at Catherine the Great’s request
• 1771 — At 64, loses sight in the left eye — total blindness
• 18 September 1783 — Dies of a cerebral hemorrhage in St. Petersburg, aged 76

Sheer Volume

• 866 papers and books in his lifetime
• The Opera Omnia began publication in 1911 and is still ongoing (over 80 volumes)
• More than half of his life’s papers were produced after he went blind (64–76)
• “Half of all mathematics was written by Euler”

Two Blindings — “Now Nothing Distracts Me”

• 1735 — Right eye lost (overwork)
• 1771 — Left eye lost (failed cataract surgery)

Euler is reported to have said, “Now I shall have fewer distractions.” After going blind he continued to work by dictation to his sons and disciples. His memory and mental arithmetic were legendary — tradition holds that he could recite Virgil’s Aeneid, all twelve books.

“Twelve years of mathematics done purely from mental computation” — under unprecedented circumstances, he still produced half his life’s output.

Euler’s Identity — “The Most Beautiful Equation”

e^(iπ) + 1 = 0

Five fundamental constants of mathematics — 0, 1, π, e, i — bound together by a single equation. Called “the most beautiful equation in mathematics” and, by some, “a proof of God.”

It is a special case (θ = π) of Euler’s formula, e^(iθ) = cos θ + i sin θ, published in the Introductio in Analysin Infinitorum (1748). Trigonometry, exponentials, and imaginary numbers are unified in a single stroke — a landmark of mathematical history.

Contributions Across Many Fields

Field	Major contributions
Analysis	Systematization of calculus, infinite series, gamma and beta functions
Number theory	FLT for n=3, two-square theorem, Euler’s totient φ(n), the zeta function
Complex analysis	Euler’s formula e^(iθ) = cos θ + i sin θ
Differential equations	Linear ODE methods, calculus of variations (with Lagrange)
Geometry	Euler line, nine-point circle, polyhedron formula V − E + F = 2
Graph theory	Königsberg bridges (1736); graph theory founded
Mechanics	Rigid-body dynamics; Euler’s equations of motion
Fluid dynamics	Euler equations
Astronomy	Lunar motion, three-body problem, orbit determination
Music theory	Theories of tuning
Optics	Refraction

The Königsberg Bridges — The Birth of Graph Theory (1736)

The problem: could the seven bridges of Königsberg be walked in a single tour? Euler proved it was impossible, and in doing so founded graph theory.

• Represent each bridge as an edge and each landmass as a vertex — a graph
• Traversability condition: zero or exactly two vertices of odd degree
• Königsberg had four vertices, all of odd degree → impossible

This paper is the origin of graph theory and topology. From a “walk over bridges” grew the lineage that leads to modern network theory and algorithms.

The Polyhedron Formula and the Euler Characteristic

For any convex polyhedron with V vertices, E edges, F faces:

V − E + F = 2

• Cube: V=8, E=12, F=6 → 8 − 12 + 6 = 2
• Tetrahedron: V=4, E=6, F=4 → 4 − 6 + 4 = 2

This is the first topological invariant — the Euler characteristic — the origin of the modern concept of “a quantity preserved under continuous deformation.”

Zeta Function and the Basel Problem

Euler studied the infinite series ζ(s) = Σ 1/n^s systematically:

• Basel problem — ζ(2) = π²/6, solved in 1734 (a decade-old open problem)
• Euler product — ζ(s) = Π_p 1/(1 − p^(-s)) (product over primes)
• Prime distribution and function theory linked — groundwork for the Riemann hypothesis

“Primes” (discrete) and “analytic functions” (continuous) meet in the zeta function — the launching point of 19th-century analytic number theory.

Euler’s Totient φ(n) — The Theoretical Foundation of RSA

The count of positive integers ≤ n coprime to n:

• φ(1) = 1, φ(2) = 1, φ(3) = 2, φ(4) = 2, φ(5) = 4 …
• Fermat–Euler theorem — when gcd(a, n) = 1, a^φ(n) ≡ 1 (mod n)
• Theoretical basis of RSA — foundational to modern cryptography

Abstract 18th-century number theory became the substrate of internet cryptography 250 years later — a classic case of delayed application of pure mathematics.

Euler’s Notation — Terms Still in Use

Modern mathematical notation was fixed by Euler:

• f(x) for a function
• e for the natural base
• i for the imaginary unit (√−1)
• π for the ratio of circumference to diameter
• Σ for summation
• sin, cos in the modern usage

All of these were fixed in Euler’s trilogy — the Introductio (1748), Institutiones Calculi Differentialis (1755), and Institutiones Calculi Integralis (1768–70) — the definitive analysis textbooks of the 18th century, read across Europe.

Education and Popular Science — Letters to a German Princess

Letters to a German Princess (1768–72) is a collection of 234 letters explaining science and philosophy to Princess Anhalt-Dessau. A general-audience science book, it became a great bestseller and was translated into many languages — an early success of “a great mathematician writing for the general public.”

Carl Friedrich Gauss — Prince of Mathematicians

Carl Friedrich Gauss (1777–1855) was a German mathematician, astronomer, and physicist. Called Princeps Mathematicorum, ranked with Archimedes and Newton among the three greatest mathematicians ever.

Number theory, algebra, geometry, analysis, astronomy, geodesy, electromagnetism, statistics — he made decisive contributions to every branch. At 19 he discovered the constructibility of the regular 17-gon; at 24 he published the Disquisitiones Arithmeticae.

Gauss the Prodigy

• Corrects his father’s arithmetic at age 3
• Age 7 — the teacher tells the class to sum 1 to 100; Gauss instantly answers 5050 (he had noticed that 1+100 = 101 repeats 50 times)
• Digests Newton, Euler, and Lagrange as a teenager
• At 19, constructs the regular 17-gon and decides on mathematics as his career

Born to a poor stonemason and gardener, Gauss caught the eye of the Duke of Brunswick, Karl Wilhelm Ferdinand, who funded his education. A scene from late-18th-century Germany in which a prince discovers and nurtures a commoner’s genius.

Life

• 30 April 1777 — Born to a poor artisan family in Brunswick
• 1788 — Duke of Brunswick recognizes his talent and awards a scholarship
• 1795–98 — University of Göttingen
• 30 March 1796 — Discovers the constructibility of the regular 17-gon at 19. Begins his mathematical diary that day
• 1799 — Dissertation on the fundamental theorem of algebra
• 1801 — Publishes Disquisitiones Arithmeticae at 24. In the same year, his orbital computation of the asteroid Ceres stuns astronomy
• 1807 — Appointed Director of the Göttingen Observatory — a post he holds for life
• 1809 — His wife Johanna dies (postpartum complications) — Gauss’s greatest grief
• 1830s — Collaborates with Wilhelm Weber on electromagnetism; invents the telegraph
• 23 February 1855 — Dies in Göttingen, aged 77

The Regular 17-gon — Breaking 2000 Years of Silence

Gauss discovered (1796) the condition for a regular n-gon to be constructible by compass and straightedge:

• n = 2^k × p_1 × p_2 × … where each p_i is a distinct Fermat prime
• Fermat primes: F_m = 2^(2^m) + 1. F_0=3, F_1=5, F_2=17, F_3=257, F_4=65537
• Consequently, the regular 3, 5, 17, 257, and 65537-gons are constructible

For 2000 years since the Greeks, no one had suspected the regular 17-gon was constructible. Gauss asked for a 17-gon carved on his tombstone; the stonemason refused, saying it would be “indistinguishable from a circle,” so the memorial shows a 17-pointed star.

Disquisitiones Arithmeticae (1801) — A Masterpiece at 24

Seven sections, published at 24. A single book that established the modern framework of number theory — an enduring classic:

• I — Theory of congruences
• II — Linear congruences
• III — Power residues
• IV — Quadratic residues; the law of quadratic reciprocity
• V — Binary quadratic forms
• VI — Applications of quadratic forms
• VII — Cyclotomic equations (theory of the 17-gon)

Quadratic Reciprocity

Gauss gave eight different proofs of this law over his lifetime. He himself called it the “golden theorem” (theorema aureum). Proving one theorem eight times is the essence of Gauss’s perfectionism.

The Fundamental Theorem of Algebra

“Every polynomial equation of degree n has n complex roots” — proved in his 1799 dissertation (age 22). Gauss gave four different proofs in his life.

The Orbit of Ceres (1801)

On 1 January 1801, Piazzi discovered a new object (later the asteroid Ceres), but observed it for only 41 days before it disappeared into the Sun’s glare. As astronomers failed to predict where to look for it again, Gauss applied least squares to compute the orbit, and by year’s end Ceres was found where Gauss predicted.

Overnight, Gauss became a hero of European astronomy — a rare case of a mathematician conquering the astronomical world.

The Gaussian (Normal) Distribution

Gauss derived the Gaussian distribution as the distribution of observational errors:

f(x) = (1 / σ√(2π)) exp(-(x − μ)² / 2σ²)

Germany’s old 10-mark note displayed his portrait alongside this curve (1989–2001) — the “distribution at the heart of statistics” made economic imagery, a rare monetary visualization of a mathematical achievement.

Perfectionism and the “Genius Who Did Not Publish”

Gauss’s motto was “Pauca, sed matura” — “few, but ripe.” He kept unfinished work unpublished. The result:

Non-Euclidean Geometry

Gauss had independently discovered it in the 1810s but feared the conservative reaction and did not publish. Lobachevsky (1829) and Bolyai (1832) got there first in print.

A surviving letter to Bolyai’s father, Farkas (a student-days friend), tells him: “I discovered this thirty years ago, but I feared the response of society and did not publish.”

Other Unpublished Work

• Elliptic functions — arrived at independently, before Abel and Jacobi
• The Gaussian integral, and much more

A mathematical diary discovered in 1898 revealed the enormity of unpublished work. “A double life of public and private mathematics” was Gauss’s distinctive strategy.

Differential Geometry and the Theorema Egregium

In Disquisitiones generales circa superficies curvas (1827), Gauss founded the intrinsic geometry of surfaces. The Theorema Egregium — the “remarkable theorem” — states that Gaussian curvature is an intrinsic property of a surface.

Riemann generalized this to manifold theory, which became the mathematical foundation of Einstein’s general relativity. Gauss’s 19th-century work on surfaces made 20th-century relativity possible.

Electromagnetism — The Telegraph with Weber

From 1831, Gauss collaborated with Wilhelm Weber in electromagnetism:

• Systematic measurement of geomagnetism
• Absolute units (the CGS Gaussian system)
• Telegraph — implemented in Göttingen in 1833 — a precursor of modern communications
• Gauss’s law — one of the fundamental laws of electromagnetism

The unit gauss (G) for magnetic flux density bears his name — the unit for the strength of magnets and magnetic disks in use today.

Gauss’s Diary

From age 19 Gauss kept a diary, recording discoveries in laconic Latin — 146 entries in total. Its discovery in 1898 transformed Gauss studies overnight. The mathematical world was stunned by the volume of unpublished results — discoveries “found but never announced” that had foreshadowed the work of Abel and Jacobi — a rare “aftermath surprise” in the history of mathematics.

Gauss’s Successors — The Göttingen Lineage

• Riemann (Gauss’s student, later professor at Göttingen) — manifold theory
• Dedekind (Gauss’s student) — real analysis
• Dirichlet (Gauss’s spiritual heir; successor at Göttingen) — analytic number theory

Gauss is the starting point of Germany’s mathematical golden age in the late 19th century. Göttingen would become “the Mecca of mathematics” through Hilbert, Klein, and Noether at the turn of the 20th century.

From This Article to the Next

We have traced the two giants of the 18th–19th centuries. The next article turns to Gauss’s contemporaries and successors — 19th-century rigorization of analysis, in Cauchy, Abel, and Jacobi. Cauchy brought ε-δ rigor to the vague foundations of calculus; the Norwegian genius Abel, who died at 26, and others carried mathematics into serious modernity.