Euclid — "A ratio is a sort of relation in respect of size between two magnitudes of the s…"
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
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"A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."
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Understanding this quote
What it means
A ratio captures how two comparable quantities relate to each other in size — not their individual values, but their proportional relationship. If one line is twice another, that relationship of twofold is the ratio. It only works between things of the same type: lengths to lengths, areas to areas. The concept strips away units and focuses purely on relative magnitude.
Relevance to Euclid
Euclid wrote this in Book V of the Elements, building on Eudoxus's theory of proportion. As a systematic geometer, Euclid needed ratios to compare incommensurable magnitudes like the diagonal and side of a square — quantities that share no common unit. His precise definition was foundational to rigorous geometric proof rather than mere intuition.
The era
Greek mathematics around 300 BCE grappled with the crisis of irrational numbers, discovered when Pythagoreans found that √2 cannot be expressed as a fraction. Euclid's careful definition of ratio sidestepped this by avoiding explicit numbers entirely, instead treating proportion geometrically — a brilliant workaround that kept Greek mathematics coherent for nearly two millennia.
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