Ratio & Proportion
A theory of when four magnitudes share 'the same ratio' — rigorous enough to handle even quantities no number could express.
Greek mathematics built a careful theory of ratio and proportion, one that defined when four magnitudes stand 'in the same ratio' — and did so even for incommensurable quantities, without leaning on numbers at all. Its rigorous form is the achievement of Eudoxus of Cnidus (4th century BCE), preserved in Book V of Euclid's Elements. This definition of equal ratios underpins all proportional reasoning in Greek geometry and anticipated the modern rigorous treatment of the real numbers.
How it traveled
- ProblemataChalcis · -322explains
- MechanicaChalcis · -322explains
- Magna MoraliaChalcis · -322explains
- FragmentaAlexandria · -300explains
- DataAlexandria · -265explains
- Sectio canonis [Sp.]Alexandria · -265explains
- Data (demonstrationes alterae)Alexandria · -265explains
- OpticaAlexandria · -265applies
- ArenariusSyracuse (Sicily) · -250explains
- Liber assumptorumSyracuse (Sicily) · -240explains
- De sphaera et cylindroSyracuse (Sicily) · -212explains
- De lineis spiralibusSyracuse (Sicily) · -212explains
- De conoidibus et sphaeroidibusSyracuse (Sicily) · -212explains
- Quadratura parabolaeSyracuse (Sicily) · -212explains
- De planorum aequilibriisSyracuse (Sicily) · -212explains
- Ad Eratosthenem methodusSyracuse (Sicily) · -212explains
- De corporibus fluitantibusSyracuse (Sicily) · -212explains
- TimaeusFormiae · -45explains
- Adversus MathematicosAlexandria · 190explains
- Opticorum recensio TheonisAlexandria · 370explains
- SynagogeAlexandriaexplains
- In Nicomachi Arithmeticam IntroductionemApameaexplains
- De utilitate mathematicaeSmyrnaexplains
- Syntaxis mathematicaAlexandriaexplains
- MetricaAlexandriaexplains
- Commentarii in libros de sphaera et cylindroAlexandriaexplains
- Scholia in Euclidis Data—explains
- Introductio arithmeticaGerasa (Jerash)explains
- Commentarius in dimensionem circuliAlexandriaexplains
- De magnitudinibus et distantiis solis et lunaeSamosexplains
- ConicaPergaexplains
- StereometricaAlexandriaexplains
- Commentarius in libros de planorum aequilibriisAlexandriaexplains
- Harmonicum enchiridionGerasa (Jerash)explains
- DioptraAlexandriaexplains
- Fragmentum [Sp.] (e cod. Paris. 453)—explains
- De Vita PythagoricaApameaexplains
- Geodaesia [Sp.]—explains
- MusicaAlexandriaexplains
- BelopoeicaAlexandriaapplies
Key passages(20)
Ad Eratosthenem methodus · Archimedes
De conoidibus et sphaeroidibus · Archimedes
De conoidibus et sphaeroidibus · Archimedes
De conoidibus et sphaeroidibus · Archimedes
De conoidibus et sphaeroidibus · Archimedes
De conoidibus et sphaeroidibus · Archimedes
De conoidibus et sphaeroidibus · Archimedes
De conoidibus et sphaeroidibus · Archimedes
De corporibus fluitantibus · Archimedes
De corporibus fluitantibus · Archimedes
De corporibus fluitantibus · Archimedes