Mathematics in Ancient Greece — Pythagoras, Euclid, Archimedes

The Moment Mathematics Became Demonstration

Babylonia and Egypt had numerical techniques. Babylonian scribes recorded Pythagorean triples like 3-4-5 on clay tablets more than a thousand years before Pythagoras, and Egyptians used a rope with knots at 3-4-5 intervals to construct right angles. But these were practical techniques, accumulations of empirical rules of thumb.

The revolution in ancient Greece was to transform them into universal laws that must be proven. “Why must this relationship hold for every right triangle?” — the invention of asking that question, and answering it by demonstration, is the central achievement of the 300 years from Pythagoras to Archimedes.

Pythagoras — All Is Number

Pythagoras (Πυθαγόρας, c. 570–495 BC) was a mathematician, philosopher, and religious teacher from the island of Samos. Ancient tradition tells of travels to Egypt and Babylonia in his youth. Around 530 BC, disliking the rule of the tyrant Polycrates on Samos, he moved to Croton in southern Italy and organized a religious and scholarly community — the Pythagorean brotherhood.

The Pythagorean Brotherhood

• Two levels — inner mathematikoi (those who learn doctrine) and outer akousmatikoi (those who keep only the rules)
• Rules — silence practice, vegetarianism, abstinence from beans (the famous unexplained puzzle), white robes
• Communal property
• Women admitted — Pythagoras’s wife Theano, his daughter Damo, and other women mathematicians
• Political activity — a base of the aristocratic-conservative faction, later persecuted by the democratic faction

Around 500 BC, an anti-Pythagorean uprising at Croton drove the community into exile. Pythagoras himself is said to have died around 495 BC at Metapontum.

“All Is Number”

The Pythagoreans held that the root of the world was number (arithmos):

• Musical harmony is expressed by ratios of string lengths (1:2 for the octave, 2:3 for the fifth, 3:4 for the fourth)
• Celestial motion is a “music of the spheres,” a numerical harmony
• Morality, justice, and friendship are numerical relations (10 is perfect, 4 is justice)

Mystical in its origin, this vision became the distant source of the later scientific worldview — nature as mathematical law.

The Pythagorean Theorem and Pythagoras’s Contribution

a² + b² = c² — in a right triangle, the square of the hypotenuse equals the sum of the squares on the other two sides.

Historically:

• Babylonians tabulated Pythagorean triples more than a thousand years earlier
• Egyptians used 3-4-5 ropes for right angles
• The Pythagoreans’ contribution was to give a general proof for any right triangle

Book I, Proposition 47 of Euclid’s Elements records the classical geometric proof. More than 400 proofs are known today.

The Discovery of the Irrational — The Crisis of √2

The Pythagorean Hippasus (5th century BC) discovered that the ratio of a square’s diagonal to its side cannot be expressed as a rational number:

Suppose √2 = p/q (reduced fraction). Squaring, p² = 2q², so p² is even, therefore p is even. Set p = 2k; then 4k² = 2q², so q² = 2k², so q is even. Both p and q are even — contradicting the assumption that the fraction was reduced.

This threatened the doctrine that “all is number (rational).” Legend says that Hippasus was thrown into the sea and killed for revealing the fact to outsiders (unreliable but symbolic). It is the first moment when mathematical truth and religious taboo collided.

Euclid — “There Is No Royal Road to Geometry”

Euclid (Εὐκλείδης, c. 325–265 BC) worked in Alexandria in Ptolemaic Egypt. His Elements (Στοιχεῖα, Stoicheia; Latin Elementa), in 13 books, remained the standard geometry textbook for over 2000 years.

Reliable biographical details are scarce. He was likely one of the scholars gathered by Ptolemy I (r. 323–283 BC) at the Museum of Alexandria. Tradition also connects him to Plato’s Academy.

Anecdotes

• When Ptolemy I asked whether there was “an easier way to learn geometry,” Euclid is said to have replied, “There is no royal road to geometry.”
• To a student who complained that geometry brought no money, he told his servant: “Give him a coin, since he must gain profit from what he learns,” and dismissed him.

Both stories come from later sources (Proclus, Stobaeus). Their historicity is not settled, but they have long stood for the authority and independence of learning.

The Elements — Summit of Mathematical Writing

The 13 books:

Book	Content
I	Plane geometry; the Pythagorean theorem (Prop. 47)
II	Geometric algebra
III	Circles
IV	Inscribed and circumscribed polygons
V	Theory of proportion (Eudoxus)
VI	Similarity
VII	Number theory basics; the Euclidean algorithm
VIII	Continued proportions
IX	Infinitude of primes; perfect numbers
X	Classification of irrationals
XI	Solid geometry basics
XII	Method of exhaustion; cone and pyramid volumes
XIII	Five Platonic solids

The Axiomatic Method

Book I opens with 23 definitions, 5 postulates, and 5 common notions. This is the origin of the style of deducing theorems from an axiomatic system.

The Five Postulates

1. A straight line may be drawn between any two points
2. A finite straight line may be extended indefinitely
3. A circle may be drawn with any center and radius
4. All right angles are equal
5. If a straight line falling on two straight lines makes the interior angles on the same side sum to less than two right angles, the two lines, extended indefinitely, meet on that side (the parallel postulate)

The fifth postulate is far more complex than the other four, and for centuries mathematicians suspected it could be derived. Only in the 19th century — through Lobachevsky, Bolyai, and Gauss — did non-Euclidean geometry show it to be genuinely independent. A revolution 2000 years in the making, planted in that fifth line.

Infinitude of Primes — A Classical Reductio

Book IX, Proposition 20 is still taught as a canonical proof by contradiction:

Suppose the primes are finite (p₁, p₂, …, pₙ). Consider N = p₁ × p₂ × … × pₙ + 1. N is not divisible by any pᵢ (remainder 1). So N contains a new prime, or is itself prime. Either way, a contradiction.

A splendid demonstration of abstract power: proving “the existence of infinity” through a finite manipulation.

Archimedes — The Greatest of the Ancients

Archimedes (Ἀρχιμήδης, c. 287–212 BC) was a mathematician, physicist, astronomer, and engineer from Syracuse in Sicily. He is called the greatest mathematician of antiquity, and is sometimes placed with Newton and Gauss as one of the three greatest of all time.

He studied in Alexandria as a young man and corresponded with Eratosthenes and Conon. Back in Syracuse he served Hieron II.

Major Works

Work	Content
Measurement of a Circle	π between 223/71 and 22/7
On the Sphere and Cylinder	Surface area and volume of the sphere; the sphere-cylinder ratio
On Spirals	Archimedean spiral; quadrature
Quadrature of the Parabola	Area of the parabola (exhaustion)
The Method	The heuristic behind quadrature
The Sand Reckoner	Notation for very large numbers; grains of sand in the universe
On Floating Bodies	The principle of buoyancy; hydrostatics
On the Equilibrium of Planes	Law of the lever; centers of gravity

Computing π

Inscribing and circumscribing 96-sided polygons in a circle, Archimedes bounded π:

• 3 + 10/71 < π < 3 + 1/7
• i.e., ≈ 3.1408 < π < 3.1429

The best estimate of the ancient world; the standard for over a thousand years.

The Method of Exhaustion — Ancestor of Integration

Begun by Eudoxus and developed by Archimedes — close in spirit to modern integration:

Inscribe and circumscribe polygons around the target curved figure. As the number of sides increases, the difference between the two shrinks arbitrarily small. The area is squeezed between the two limits.

Archimedes computed the area of the parabola, the volume of the sphere, the area of the spiral, and more — a preview of the calculus Newton and Leibniz would create 2000 years later.

“Eureka” and “Do Not Disturb My Circles”

Discovery of Buoyancy

Hieron asked Archimedes whether a goldsmith had substituted silver into his crown — without damaging the crown. Watching bathwater overflow, Archimedes reached the principle of buoyancy. He is said to have run naked through the streets shouting “Eureka! (I have found it)” (Vitruvius).

The truth is debated, but the story survives as the archetypal moment of discovery.

His Death

In 214–212 BC, the Roman army besieged Syracuse. Archimedes fought back with catapults, giant claws, and heliostat mirrors. In 212 BC, Syracuse fell.

At the moment of the fall, Archimedes was absorbed in a geometrical diagram drawn in the sand. When a Roman soldier burst in, he is said to have shouted “Do not disturb my circles” (Noli turbare circulos meos). The commander Marcellus had ordered him spared, but a soldier who did not know this killed him.

Marcellus mourned him and had a sphere inscribed in a cylinder carved on his tomb. About 130 years later, Cicero, as quaestor at Syracuse, rediscovered the tomb and made it known to Rome — a Roman politician tracking down the grave of a mathematician, a scene from the Mediterranean world.

The Archimedes Palimpsest

The Method was long thought lost, until in 1906 the Archimedes Palimpsest — a prayer book overwritten atop an earlier manuscript — was discovered in Istanbul and recovered. It showed that Archimedes had used infinitesimals in his heuristic calculations, and prompted a fresh appreciation of just how modern his methods had been.

From This Article to the Next

We have traced the 300 years in which “mathematics as demonstration” was established in ancient Greece. The next article covers the interlude after Archimedes — medieval and early-modern mathematics. The algebraic seed of Diophantus (3rd century), al-Khwārizmī in the 9th century, Fibonacci in the 12th, and, on an independent line, 17th-century Japan’s Seki Takakazu and wasan. A 1500-year story in which the Islamic world and East Asia filled the “European vacuum.”