Method of exhaustion and theory of proportions

What was discovered

Rigorous technique for calculating areas and volumes by approximating them with inscribed polygons and using double reduction to absurdity, together with a definition of equal ratios valid for incommensurable magnitudes. It is the direct forerunner of integral calculus and the construction of the real numbers.

Recorded in Books V and XII of Euclid's Elements.

What it led to

• Archimedes's calculation of areas
• integral calculus
• Dedekind cuts

Built on

• Zeno paradoxesc. 450 BCE
• irrationality of the square root of 2

Made possible

• Euclid Elements and the axiomatic methodc. 300 BCE
• Quadrature of the parabola and The Methodc. 250 BCE
• Archimedes computation of pic. 250 BCE
• Al-Quhi's perfect compassc. 980
• Mean speed theorem and Oresme graphsc. 1335
• Dedekind cuts and ideals1872

Links