Liu Hui's π Algorithm - Quick Method

Quick Method

Calculation of square roots of irrational numbers was not an easy task in the third century with counting rods. Liu Hui discovered a short cut by comparing the area differentials of polygons, and found that the proportion of the difference in area of successive order polygons was approximately 1/4.

Let D_N denote the difference in areas of N-gon and (N/2)-gon

He found:

1

Hence:

Area of unit radius circle =

In which

That is all the subsequent excess areas add up amount to one third of the

area of unit circle2

Liu Hui was quite happy with this result because he had acquired the same result with the calculation for a 1536-gon, obtaining the area of a 3072-gon. This explains four questions:

1. Why he stopped short at A₁₉₂ in his presentation of his algorithm. Because he discovered a quick method of improving the accuracy of π, achieving same result of 1536-gon with only 96-gon. After all calculation of square roots was not a simple task with rod calculus. With the quick method, he only needed to perform one more subtraction, one more division (by 3) and one more addition, instead of four more square root extractions.
2. Why he preferred to calculate π through calculation of areas instead of circumferences of successive polygons, because the quick method required information about the difference in areas of successive polygons.
3. Who was the true author of the paragraph containing calculation of
4. That famous paragraph began with "A Han dynasty bronze container in the military warehouse of Jin dynasty....". Many scholars, among them Yoshio Mikami and Joseph Needham, believed that the "Han dynasty bronze container" paragraph was the work of Liu Hui and not Zu Chongzhi as other believed, because of the strong correlation of the two methods through area calculation, and because there was not a single word mentioning Zu's 3.1415926 < π < 3.1415927 result obtained through 12288-gon.