How did Eudoxus influence Books V and XII of Euclid’s Elements?

Eudoxus of Cnidus significantly influenced the Elements by providing the mathematical foundation for the general theory of proportions and the rigorous treatment of limits.

Book V: Theory of Proportions

Eudoxus is credited with the general account of proportions found in Book V. Before his work, proofs regarding proportions required separate treatments for different types of magnitudes, such as lines, surfaces, and solids. Eudoxus revolutionized this by providing general proofs that applied to all magnitudes. This theory, which addresses incommensurable magnitudes, is essential for the logical rigor of the Elements and served as a geometric theory of numbers for centuries.12

Book XII: Method of Exhaustion

Book XII incorporates Eudoxus’s method of exhaustion, a precursor to modern integration, to handle areas and volumes of curved figures. Key contributions include:12
* Proving that the volumes of pyramids and cones are one-third those of prisms and cylinders with the same bases and heights.1
* Demonstrating that the areas of circles are proportional to the squares of their diameters.12
* Establishing that the volumes of spheres are proportional to the cubes of their diameters.2