mathematics
Incommensurable Magnitudes
ἀσύμμετρα
Some lengths share no common measuring stick at all — a discovery that shattered the faith that everything is number.
Two magnitudes are incommensurable when no single unit, however tiny, measures both of them a whole number of times. The classic case is the side and diagonal of a square. Tradition credits the discovery to the Pythagorean school of the 5th century BCE, with legend naming Hippasus, and Aristotle preserves the proof that the diagonal is 'irrational.' The finding forced Greek mathematics to treat such quantities geometrically rather than as numbers — a problem that Eudoxus' theory of proportion later tamed and that Euclid's Book X classified exhaustively.
Where this idea shows up
3 Greek sourcesWhere to read it
- Fragments & Testimonia300 BCEEuclid· Alexandria
- De planorum aequilibriis212 BCEArchimedes· Syracuse (Sicily)
- Adversus Mathematicos190 CESextus Empiricus· Alexandria