Method of Exhaustion
E620682
The Method of Exhaustion is an ancient Greek technique, developed notably by Eudoxus and used by Archimedes, for finding areas and volumes by inscribing and circumscribing sequences of shapes that increasingly approximate a figure, anticipating the principles of integral calculus.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
ancient Greek mathematics technique
ⓘ
mathematical method ⓘ |
| anticipates |
Riemann integration
NERFINISHED
ⓘ
integral calculus ⓘ limit concept ⓘ |
| appliedBy |
Archimedes in Measurement of a Circle
NERFINISHED
ⓘ
Archimedes in On the Quadrature of the Parabola ⓘ Archimedes in On the Sphere and Cylinder NERFINISHED ⓘ |
| approach |
circumscribing polygons about a circle
ⓘ
inscribing polygons in a circle ⓘ |
| basedOn |
Eudoxus theory of proportion
NERFINISHED
ⓘ
axiom of Eudoxus NERFINISHED ⓘ |
| contrastWith | heuristic infinitesimal methods ⓘ |
| coreIdea |
approximating areas and volumes by sequences of circumscribed figures
ⓘ
approximating areas and volumes by sequences of inscribed figures ⓘ using limiting processes without explicit limits ⓘ |
| developedBy | Eudoxus of Cnidus NERFINISHED ⓘ |
| era | classical Greek mathematics ⓘ |
| field |
geometry
ⓘ
mathematical analysis ⓘ |
| formalNature | rigorous geometric method ⓘ |
| goal | eliminate any remaining difference between figure and approximating shapes ⓘ |
| historicalPrecursorOf |
infinitesimal calculus
NERFINISHED
ⓘ
method of limits NERFINISHED ⓘ |
| influenceOn |
early modern calculus developers
ⓘ
later Hellenistic mathematics ⓘ |
| logicalStructure |
assumes difference between quantity and approximating sequence
ⓘ
derives contradiction when difference is assumed positive ⓘ |
| methodType |
indirect proof
ⓘ
reductio ad absurdum ⓘ |
| philosophicalBasis |
rejection of actual infinitesimals
ⓘ
use of potential infinity via successive refinement ⓘ |
| relatedConcept |
Archimedean property
NERFINISHED
ⓘ
Eudoxian proportion theory NERFINISHED ⓘ geometric series approximation ⓘ |
| timePeriod | ancient Greece NERFINISHED ⓘ |
| usedBy | Archimedes NERFINISHED ⓘ |
| usedFor |
finding area of plane figures
ⓘ
finding volume of solids ⓘ proving area formulas rigorously ⓘ proving volume formulas rigorously ⓘ |
| usedToProve |
area of a circle equals pi times radius squared
ⓘ
volume of a cone equals one third base times height ⓘ volume of a sphere equals four thirds pi times radius cubed ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.