The Birth of Greek Geometry

From the Pythagorean theorem, through the shock of irrational numbers, to Euclid's Elements installing proof itself as a universal form.

About 25 min·8 nodes

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The path across the map

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STEP 1 · ConceptCrotone· Discovery

Pythagorean Theorem

In a right triangle, the square of the hypotenuse equals the sum of the other two squares. Related calculations and proofs appear in Babylonian, Indian, Chinese, and Greek sources.

↓Next step

STEP 2 · Mathematician~BCE 600Miletus· Main activity

Thales of Miletus

An ancient starting point that must be read carefully across later sources. No writing by Thales survives, but Aristotle reports that he sought the origin of things in water — one of the earliest Greek attempts to explain nature through a common principle rather than divine whim alone. Herodotus says Thales foretold the year of the eclipse of 585 BCE and that the Medes and Lydians stopped fighting after it. Whether anyone then could predict a solar eclipse at a particular place is disputed; modern historians also consider a rough guess or later attribution. Stories about measuring a pyramid and proving geometric theorems were recorded centuries later. Learning to separate these levels of certainty is part of what makes Thales compelling.

↓70 years later

STEP 3 · Mathematician~BCE 530Crotone· Main activity

Pythagoras

Remembered through the phrase all is number, Pythagoras founded a community at Crotone that combined mathematics, music, astronomy, ethics, and rules of life. No writing by him survives, and later Pythagorean sources often attribute a community’s work to its founder, so individual discoveries are difficult to assign. Musical ratios, the organization of results about right triangles, and debates over incommensurable magnitudes all belong to this wider tradition. Stories about forbidden beans or Hippasus drowning appear in later, conflicting forms and should be read as traditions rather than eyewitness history.

↓80 years later

STEP 4 · Mathematician~BCE 450Crotone· Main activity

Hippasus of Metapontum

A Pythagorean figure surrounded by more layers of tradition than secure fact. Hippasus is associated with Metapontum and the fifth-century BCE Pythagorean movement. Much later sources remember someone who disclosed a mathematical secret, but disagree over whether the secret concerned incommensurable magnitudes or the construction of a dodecahedron, and whether the punishment was expulsion, shipwreck, or divine drowning. No ancient author explicitly says that Hippasus was the first to prove the irrationality of √2. It is therefore more accurate to treat him as the symbolic name attached to the Pythagorean encounter with lengths no fraction can express, not as a certain lone discoverer.

↓Same year

STEP 5 · Number system~BCE 450Crotone· Discovery

Completion of the Reals

The complete ordered field obtained by filling every gap in the rationals, joining rational and irrational numbers into one unbroken line.

↓150-year span

STEP 6 · Mathematician~BCE 300Alexandria· Main activity

Euclid

Author of The Elements. He established the foundations of mathematics through an axiomatic system that defined rigor for 2,300 years.

↓50 years later

STEP 7 · Mathematician~BCE 250Syracuse· Main activity

Archimedes

The surviving works are more remarkable than the legends. Archimedes used exhaustion and mechanical reasoning to study spheres, cylinders, parabolic areas, and bounds for π; his results anticipate parts of later integration without being modern calculus. His work on floating bodies survives, while the bath, burning-mirror, and last-words stories come from much later accounts. The rediscovered Archimedes Palimpsest revealed more of how he reasoned.

↓Next step

STEP 8 · RiverAthens· Journey start

The River of Euclidean Geometry

Euclid's Elements traveling from Hellenistic Alexandria via Arabic translation through Baghdad and Toledo to medieval European universities.

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