Liu Hui
Life
A mathematician of third-century Cao Wei China. Little is known about his life, but his commentary on the Nine Chapters on the Mathematical Art, dated around 263, and the Sea Island Mathematical Manual survive. He added explanations of why many procedures work, treating systems of linear equations with positive and negative numbers, areas and volumes, and right-triangle relations. By repeatedly increasing the number of sides of an inscribed regular polygon, he obtained π ≈ 3.1416 and examined the accuracy of approximations. The nine problems of the Sea Island Mathematical Manual use observations from two positions and right-triangle relations to determine inaccessible heights and distances.
Decisive moments
Commentary on the Nine Chapters — explaining procedures
He added explanations to procedures involving fractions, ratios, systems of equations, areas, volumes, and right triangles, and examined approximation accuracy and gaps in some arguments.
Approximating π with regular polygons
By repeatedly doubling the sides of an inscribed regular polygon, Liu Hui refined the approximation of π to 3.1416. The method combines the idea of polygons approaching a circle with checks on approximation error.
Sea Island Mathematical Manual — 9 ways to measure the inaccessible
Nine problems use observations from two positions and right-triangle relations to determine values that are hard to measure directly, such as an island’s height, a valley’s depth, or a river’s width.
If this person hadn't existed
This is a thought experiment about influence, not a verified historical fact.
Without Liu Hui’s commentary, an important record explaining and testing procedures in the Nine Chapters would have been lost. It would be too strong to say that all Chinese mathematics would therefore have lacked reasoning, but the route by which later scholars studied and extended this classic might have differed.
Beyond MathVoyage
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