Incommensurable Magnitudes
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.
How it traveled
- SynagogeAlexandriaexplains
- In Aristotelis Analyticorum Priorum Librum I CommentariumAthensexplains
Key passages(13)
In Aristotelis Analyticorum Priorum Librum I Commentarium · Alexander of Aphrodisias
In Nicomachi Arithmeticam Introductionem · Iamblichus
De utilitate mathematicae · Theon Smyrnaeus
In Aristotelis Analyticorum Priorum Librum I Commentarium · Alexander of Aphrodisias
De planorum aequilibriis · Archimedes
Adversus Mathematicos · Sextus Empiricus
In Platonis Rem Publicam Commentarii · Proclus
In Aristotelis analytica priora [Sp.] · Pseudo-Ammonius