Liu Hui's π algorithm: Difference between revisions

Revision as of 16:22, 19 March 2021

Revision as of 01:03, 26 May 2021

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Liu Hui argued:

:"''Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the finer we cut, the smaller the loss with respect to the area of circle, thus with further cut after cut, the area of the resulting polygon will coincide and become one with the circle; there will be no loss''".th

Apparently Liu Hui had already mastered the concept of the limit<ref>First noted by Japanese mathematician [[Yoshio Mikami]]</ref>

: <math>\lim_{N \to \infty}\text{area of }N\text{-gon} = \text{area of circle}. \, </math>

Further, Liu Hui proved that the area of a circle is half of its circumference multiplied by its radius. He said: