What theory of ratios and proportions appears in Book V of Euclid’s Elements?

Book V of Euclid’s Elements presents a general theory of ratios and proportions that is widely attributed to the Greek mathematician Eudoxus of Cnidus.12

This theory was revolutionary because it provided a rigorous mathematical framework that accounted for incommensurable magnitudes—quantities that cannot be expressed as a ratio of two integers, now known as irrational numbers. Before this development, mathematical proofs regarding proportions often required separate, specific treatments for lines, surfaces, and solids. Eudoxus’s approach allowed for general proofs that applied to all magnitudes of the same sort, forming the foundation for a geometric theory of numbers that remained authoritative until the development of analytic theory in the late 19th century.132