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## Problem

There is another volume measuring 1,644,866,437,500 cubic chi. What is the diameter of the sphere?

## Solution

Spherical extraction method: Take the volume, multiply by 16, and divide by 9. Extract the cube root to obtain the diameter of the ball.

This solution found in the Jiuzhang Suanshu is not accurate. The formula used here is

.

This meant the diameter was taken as the cube root of 2,924,207,000,000. This value turns out to be 14300.

However the Chinese did discover the precise formula of the volume of the sphere

.

Using this formula the correct solution (rounded to the nearest whole number) is

.

Yet one question remains. How did one calculate the cube root of a large number like 2,924,207,000,000? The answer lies in scaling and an iterative algorithm for extracting cube roots. Here I present the algorithm with algebraic equations. The algorithm employed in ancient China two millennia ago was reduced to clever computations that replicate these equations (more specifically the coefficients of these equations).

Initialization

We need to solve . Notice the six zeroes at the end?

Let . This will help reduce the number to cube root.

Iteration 1

Make the first estimate for determining the error bound .

Let , then

Iteration 2

Since , update the substitution .

Iteration 3

Since , we may end the procedure.

Therefore, if , then

.