mathematics
Division in Extreme & Mean Ratio
ἄκρος καὶ μέσος λόγος
Cut a line so the whole stands to the larger piece exactly as the larger piece stands to the smaller.
A line is divided 'in extreme and mean ratio' when the whole is to its greater segment as that segment is to the lesser. This is the proportion later nicknamed the golden section. Euclid of Alexandria (c. 300 BCE) defines and constructs it in the Elements, where it is the key to building the regular pentagon along with the dodecahedron and icosahedron. The familiar names 'golden ratio' and φ are modern labels from the 19th and 20th centuries, not Greek terms at all.
Where this idea shows up
3 Greek sourcesWhere to read it
- Liber assumptorum240 BCEArchimedes· Syracuse (Sicily)
- De sphaera et cylindro212 BCEArchimedes· Syracuse (Sicily)