Archimedes & Apollonius
The greatest mathematician of antiquity and the master of conic sections
5.1 Archimedes of Syracuse
Archimedes (c. 287β212 BCE) is widely regarded as the greatest mathematician of the ancient world β and one of the greatest of all time. Working in Syracuse, Sicily, he made profound contributions to geometry, mechanics, hydrostatics, and mathematical physics.
Using the method of exhaustion β a precursor of integral calculus β Archimedes computed the area under a parabola, the surface area and volume of a sphere, and showed that the area of a circle equals ΟrΒ². He approximated Ο by inscribing and circumscribing regular polygons, obtaining 223/71 < Ο < 22/7.
The Method (Lost and Found)
In 1906, a palimpsest discovered in Constantinople revealed Archimedes's lost work The Method, in which he described how he actually discovered his results β by imagining shapes as composed of infinitely thin slices, a technique strikingly close to modern integration. He considered this approach heuristic and published only his rigorous proofs using exhaustion.
5.2 Apollonius and the Conics
Apollonius of Perga (c. 262β190 BCE) wrote the definitive ancient treatise on conic sections β the curves obtained by slicing a cone with a plane. He gave them the names we still use: ellipse, parabola, and hyperbola.
Apollonius's Conics (eight books, of which seven survive) developed the theory with extraordinary thoroughness and elegance. Nearly two thousand years later, Kepler discovered that planets move in ellipses, and Newton showed that all conic sections arise as possible orbital paths under inverse-square gravitational force β vindicating Apollonius's purely geometric investigation.
Video Documentary
Watch the documentary on Apollonius in our Video Lectures page β βApollonius of Perga: The Father of Conics Who Inspired Newton and Kepler.β
5.3 Key Achievements
Archimedes
- β’ Area of a parabolic segment
- β’ Surface area and volume of a sphere
- β’ Approximation of Ο (3.1408... to 3.1429...)
- β’ The Archimedean spiral
- β’ Principle of buoyancy (Eureka!)
- β’ The lever and mechanical advantage
Apollonius
- β’ Named ellipse, parabola, hyperbola
- β’ Focal properties of conics
- β’ Tangent lines to conics
- β’ Normal lines and curvature
- β’ Foundation for planetary orbits (Kepler)