Book 12. Poster. Proportional stereometry. Euclid’s Elements

Book 12 Poster: Proportional Stereometry | Euclid's Elements A2 Print

The method of exhaustion applied to solid figures—Book XII explores volumes and proportions of three-dimensional objects including pyramids, cones, cylinders, and spheres. This A2 poster visualizes Euclid's use of Eudoxus's method of exhaustion, a precursor to integral calculus.

Ancient Calculus: The Method of Exhaustion

Book XII of Euclid's Elements applies the method of exhaustion to prove theorems about the volumes of solid figures. This technique—inscribing and circumscribing polygons with increasing numbers of sides—anticipates integral calculus by nearly 2,000 years. Part of Kronecker Wallis's completion, this poster explores:

• Circles are to one another as the squares of their diameters
• Volumes of pyramids with the same base and height are equal
• Volumes of pyramids are proportional to their bases
• Volumes of cones follow the same principles as pyramids
• Volumes of cylinders are proportional to their bases
• Spheres are to one another as the cubes of their diameters (Proposition 18)

These 18 propositions represent some of the most sophisticated mathematical reasoning in ancient geometry, using rigorous logic to establish results about curved surfaces and volumes without the benefit of algebraic notation or modern calculus.

Precursor to Modern Calculus

Second solid geometry book, building on Book 11. Part of the complete thirteen-book visual edition. Prepares for Book 13's Platonic solids. Complete your collection with the 14-poster set.

Specifications:

• Size: A2 (420 × 594 mm / 16.5 × 23.4 inches)
• Subject: Proportional Stereometry and Volumes
• Unframed, premium archival printing
• Visualizing ancient proto-calculus techniques