What the work claims
Liu Hui claimed to prove foundational geometric theorems, compute volumes of major solids, improve π beyond Archimedes and Ptolemy, and systematise solutions to linear equations — all within commentaries on The Nine Chapters on the Mathematical Art and Haidao Suanjing.
How it was done
Liu Hui used geometric dissection, polygonal approximation, and a systematic algorithmic method — including what is now called Gaussian elimination — to derive results in geometry and arithmetic.
What holds up
His proof of a theorem identical to the Pythagorean theorem holds. His volume calculations for cone, cylinder, frustum, prism, pyramid, tetrahedron, and wedge hold. His π approximation of 3.14159 using a 3072-sided polygon holds.
What does not
He failed to compute the volume of a sphere. He explicitly acknowledged this gap and deferred it to future mathematicians.
Why it matters beyond the lab
It matters because it refutes the Eurocentric narrative that algorithmic rigour and geometric proof emerged only in Greece or Renaissance Europe. Liu Hui’s methods were operational, reproducible, and embedded in applied surveying — not abstract speculation.
Is it worth your time
Yes. His work demonstrates that rigorous proof, numerical approximation, and algorithmic problem-solving were fully developed in 3rd-century China — independently of Greek or later European traditions.