Limiting and accumulation in classical Chinese mathematics

Equal cross-sections,equal volumes.

There was no calculus notation and no modern definition of a limit. Begin with the operations themselves: make a cut, reduce what is lost, and let the result emerge.

Make the first cut ↓

Four operations
cut · approach · accumulate · connect
Watch a mathematical intuition
take shape through action.

Do not replace old terms with modern onesOperate first:see how an idea grows from a procedure

I · Cut

割

“割之又割，以至于不可割，则与圆周合体而无所失矣。”Liu Hui, commentary on the Nine Chapters: “Cut again and again… until it coincides with the circle and nothing is lost.”

Begin with a regular hexagon inscribed in a circle and double the number of sides. The polygon never becomes a circle, yet it draws ever closer.

circle

Sides6

Approximation of π3.000000

Uncovered area17.301%

6 sidesDrag to keep doubling3,072 sides

The values use modern trigonometry to reconstruct Liu Hui’s geometric procedure for a circle of radius 1.

II · Approach

逼

“割之弥细，所失弥少。”Liu Hui: “The finer the cutting, the less is lost.”

Bring the error into view. Each doubling rapidly tightens the interval that contains π.

Inscribed lower boundπCircumscribed upper bound

3.000000π3.464102

Undetermined interval0.464102

Visible gapContinue approachingBarely visible

Both bounds move; the target remains between them.

III · Accumulate

积

“幂势既同，则积不容异。”Zu Geng, quoted in Tang commentaries: “If cross-sectional areas and heights agree, the volumes cannot differ.”

Here mi refers to cross-sectional area and shi to height. Slice a sphere into thin layers: their cross-sections together determine the whole volume.

Slices4

Thickness0.5000 r

Volume estimate4.31969

Relative error3.125%

ThickDrag the slice thicknessThin

The transparent stack is a modern visualization. Zu Geng’s principle compared equal-height cross-sections; it did not state a Riemann sum.

IV · Connect

合

Place the old operations beside modern notation. Drag the seam to see what structural intuition continues—and what formal language came later.

Classical procedural language

割6 → 12 → 24 sides

逼finer cutting, less loss

积equal sections imply equal volume

Modern formal language

partitionn → ∞

limitlim A_n = πr²

integralΣ A(zᵢ)Δz → ∫ A(z) dz

合

Classical operationMove the boundaryModern form

Liu Hui and Zu Geng did not have modern functions, formal limits, or integral signs. They left exceptionally clear procedures for partition, approximation, and reasoning from equal cross-sections.

Texts and notes

Begin with the sources

Nine Chapters with Liu Hui’s commentaryChinese Text Project Liu Hui’s circle procedureChinese original at Wikisource Nine Chapters, Scroll FiveSolid dissection and “the finest is called minute” Zu Geng’s principle and the sphereHistory of mathematics, NTU

The animations use modern numerical methods to expose historical ideas. They do not imply that the source texts used functions, coordinates, limit notation, or Riemann integration.