Summary
Archimedes’ "On the Sphere and Cylinder" establishes geometrical methods for calculating the dimensions of spheres, cones, and cylinders, demonstrating that the volume of a sphere is two-thirds the volume of its circumscribing cylinder. Book one focuses on the surface areas of right cylinders and cones, and the surface area and volume of spheres and spherical segments. Book two addresses problems related to segments of spheres, including how to divide a sphere into segments with a given ratio of surfaces or volumes, and how to construct a segment with a specific volume or surface area relative to another.
The treatise, highly valued by Archimedes, employed a refined method of exhaustion, equivalent to integration, to achieve its results. Readers learn Archimedes' precise geometrical proofs for key relationships between geometric solids, including the groundbreaking comparison of a sphere's volume to that of its circumscribing cylinder, a discovery so significant it was inscribed on his tomb.
Key concepts
- Surface of right cylinder — Calculations for the surface area of a right cylinder.
- Surface of right cone — Calculations for the surface area of a right cone.
- Surface of sphere — Calculations for the surface area of a sphere.
- Volume of sphere — Calculation of the volume of a sphere and its relation to a circumscribing cylinder.
- Segment of sphere — Problems involving dividing a sphere into segments with specific ratios of surfaces or volumes.
- Solid rhombi — Geometrical dimensions calculated for a shape described as a "solid rhombus."
From the book
Title: On the Sphere and Cylinder by Archimedes← Archimandrite 1911 Encyclopædia Britannica , Volume 2 Archimedes by Thomas Little Heath Archimedes, Screw of → See also Archimedes on Wikipedia ; and our 1911 Encyclopædia Britannica disclaimer . 359765 1911 Encyclopædia Britannica , Volume 2 — Archimedes Thomas Little Heath ARCHIMEDES ( c. 287–212 B.C. ), Greek mathematician and inventor, was born at Syracuse, in Sicily. He was the son of Pheidias, an astronomer, and was on intimate terms with, if not related to, Hiero, king of Syracuse, and Gelo his son. He studied at Alexandria and doubtless met there Conon of Samos, whom he admired as a mathematician and cherished as a friend, and to whom he was in the habit of communicating his discoveries before publication. On his return to his…
Popular questions readers ask
- Archimedes explicitly valued mathematical research over his famous mechanical contrivances. If you were explaining his perspective, why might someone consider practical, world-changing inventions "beneath the dignity of pure science," and what does this reveal about his ultimate intellectual priorities?
- Describe Archimedes' "Eureka!" discovery regarding Hiero's crown in terms a curious 10-year-old could understand. Beyond the simple explanation, what fundamental principle did he uncover, and why was it so significant that it laid the "foundation of that whole science" of hydrostatics?
- How do the "Eureka!" story (hydrostatics) and the "Give me a place to stand and I will move the earth" statement (mechanics) both illustrate Archimedes' unique approach to understanding and leveraging physical principles, even though they concern different domains of science?
- The text notes that the story of the burning mirror is "discredited" by historians. What is the value of critically evaluating popular stories against historical accounts when studying a figure like Archimedes, and what does this process tell us about the nature of scientific and historical truth?
- How did Archimedes' personal relationships, particularly with King Hiero and the mathematician Conon of Samos, likely influence not only the types of problems he investigated but also the dissemination and eventual legacy of his discoveries?