The Greek Who Made Geometry Honest

Eudoxus of Cnidus was born around 408 BC in the small Greek city of Cnidus on the southwestern coast of Anatolia and died around 355 BC, almost certainly in the same city where he had been born. He spent most of his adult life moving between Cnidus, Athens (where he studied at Plato’s Academy and later led a satellite school of his own), Cyzicus on the southern shore of the Black Sea, and Heliopolis in Egypt, where he spent over a year studying with the Egyptian priestly astronomers. He was, by every available account, the most important Greek mathematician of his century — and the century in question included Plato, Aristotle, Theaetetus, and Archytas.

He left no surviving writings. Everything we know about his mathematical work comes from later authors — most importantly Euclid (whose Elements, completed approximately fifty years after Eudoxus’s death, preserves what is generally thought to be a substantially complete Eudoxean treatment of proportion theory in Book V), Archimedes (whose work on geometric volumes used the Eudoxean proof technique two centuries later), and Aristotle (who refers to Eudoxus’s astronomical model in Metaphysics).

What he invented — and what makes him one of the most consequential mathematicians in human history — is the rigorous technique that let Greek geometry handle curves.

What was wrong with geometry before Eudoxus

Greek mathematics in the early fourth century BC had a problem. The geometrical results that involved only straight lines and simple polygons — the kind of theorems Euclid’s Elements would eventually compile in Books I through IV — could be proven by direct construction. You could show that the angles of a triangle add up to two right angles by drawing a particular parallel line and identifying alternate angles. The proof was finite, mechanical, and complete.

The geometrical results that involved curves — the area of a circle, the volume of a cone, the surface area of a sphere — could not be proven the same way. The Greek mathematicians who had attempted these problems (Hippocrates of Chios, Antiphon, Bryson) had been forced to use intuitive arguments that worked by exhaustion in the loose sense: “as you inscribe more and more polygons in a circle, the polygons get closer and closer to the circle’s area.” This kind of argument was suggestive but not rigorous in the sense Greek mathematicians of the Eudoxean period wanted. It depended on an intuitive grasp of limits — of what happens “in the limit” as you take more and more steps — that the surviving mathematical vocabulary could not formalize.

The reason the formalization was difficult is fundamentally philosophical. Greek mathematics in the period had committed itself, partly under Platonic influence and partly for internal logical reasons, to the principle that magnitudes (lengths, areas, volumes) were not numbers — they did not have the cleanly additive properties of integers — and that proofs about them therefore could not appeal to numerical limits. A proof about a curved area had to be a proof about the area itself, expressed entirely in geometrical language, with no recourse to “as n approaches infinity.”

The breakthrough — the technique that became known as the method of exhaustion and was the central proof technique of advanced Greek geometry for the next six centuries — was Eudoxus’s.

What the method actually says

The method of exhaustion is, in the modern phrasing, a proof by contradiction using inscribed and circumscribed approximations. The structure is as follows.

To prove that the area of a particular curved shape A equals a particular value V, the mathematician constructs two sequences of approximating polygons: an inscribed sequence that fits inside the curve, and a circumscribed sequence that contains the curve. The inscribed sequence has areas approaching V from below; the circumscribed sequence has areas approaching V from above. The mathematician then shows:

• If the area of A were greater than V, then for some sufficiently fine inscribed polygon, the polygon’s area would also exceed V — which would contradict the polygon being inscribed in A (since the inscribed polygon must be smaller than A but we just claimed the polygon’s area is greater than V, which is the area we said A equals, contradiction).
• If the area of A were less than V, then for some sufficiently fine circumscribed polygon, the polygon’s area would also be less than V — which would similarly contradict the polygon being circumscribed.

Since A’s area cannot be greater than V and cannot be less than V, it must equal V.

The structure is elegant. It avoids the philosophical problem of “infinite limits” by treating the approximating polygons as a sequence of finite constructions and the equality of the curved area to V as a logical consequence of the impossibility of inequality. The proof never says “the polygon becomes the curve at infinity”; it says “for any small difference, you can find a polygon within that difference of the curve, therefore there can be no difference at all.”

In modern mathematical terms, this is essentially a proof using the definition of a limit — and is structurally what every modern calculus textbook does when proving statements about integrals. The Eudoxean argument and the modern ε-δ argument are logically identical. Eudoxus’s version is two thousand two hundred years older.

What was built on it

The most spectacular applications were Archimedes’s. Two centuries after Eudoxus, Archimedes used the method to prove the area of a parabolic segment, the volume of a sphere, the volume of a cone, the surface area of a sphere, and several dozen other curved-figure theorems that constitute the bulk of his geometrical work. The method appears as the standard proof structure throughout Archimedes’s surviving treatises — and the discovery process he used (mechanical balance arguments) was developed privately, then converted to the public Eudoxean format for publication. Archimedes thought the Eudoxean format was the only rigorous form a geometrical argument could take.

Euclid’s Elements Book V — the foundational text of Greek proportion theory — is generally considered to be a substantially Eudoxean treatment. The definitions of ratio and proportion given there, which let Greek mathematics handle the comparison of incommensurable magnitudes (the diagonal and side of a square, for example), are the conceptual scaffolding that made the method of exhaustion possible. The same definitions appear in modern real analysis textbooks as the basis for the formal construction of the real number line.

The method of exhaustion remained the standard rigorous-proof technique for curved-figure theorems until the development of formal calculus by Newton and Leibniz in the 1670s and 1680s. Newton’s and Leibniz’s techniques were faster and more general, but they had a famous logical problem (the “fluxions” or “infinitesimals” that were used in the derivations were of uncertain logical status) that took another two centuries to fully resolve. The resolution, when it came in the late nineteenth century via Cauchy and Weierstrass, was — in its final epsilon-delta form — essentially the Eudoxean argument made systematic.

Greek geometry’s commitment to rigorous proof of statements about curves, which is now a permanent feature of modern mathematical method, began with one man, in the Aegean, in the middle of the fourth century BC.

He is not buried at Cnidus. His grave has not been found. The site of the school he ran in Athens, near the Academy, is not known. There is a small bronze plaque in his honor in the modern town of Datça in southwestern Turkey, near the ruins of Cnidus, installed by the Turkish ministry of culture in 1986. The plaque is in Turkish. It gives his dates approximately and names him as “a mathematician of ancient Cnidus.” It does not explain what he did.