Division in Extreme & Mean Ratio
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.
How it traveled
- FragmentaAlexandria · -300explains
- De sphaera et cylindroSyracuse (Sicily) · -212explains
- SynagogeAlexandriaexplains
- Commentarii in libros de sphaera et cylindroAlexandriaexplains
- De utilitate mathematicaeSmyrnaexplains
- BelopoeicaAlexandriaexplains
Key passages(20)
Commentarii in libros de sphaera et cylindro · Eutocius
Commentarii in libros de sphaera et cylindro · Eutocius
Commentarii in libros de sphaera et cylindro · Eutocius
Commentarii in libros de sphaera et cylindro · Eutocius
Commentarii in libros de sphaera et cylindro · Eutocius
Mechanicorum fragmenta · Hero of Alexandria