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Concept

Archimedes

Archimedes of Syracuse, whose method of exhaustion pinned a curved quantity between two sequences of straight-sided ones and ruled out every other value. He proved the disc's area, the sphere's volume and the parabolic segment that way, and computed the first good bounds on pi from ninety-six-sided polygons.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

A circle unrolled into a triangle

Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.

geometry · Circle area A circle trapped between two 12-sided polygons. A circle with a regular polygon of 12 sides inscribed in it and another circumscribed about it, beside a table of the bounds on pi obtained by doubling the side count.

Pinned between two sequences

The ring dissection makes the answer obvious and proves nothing. Archimedes' method proves it and makes nothing obvious — it never exhibits the area at all, it rules out every other value — and the recursion that drives it computes π by hand with one square root a step.

geometry · Circle area A hemisphere and a cylinder with a cone taken out, sliced at one height. Two solids drawn in profile — a hemisphere, and a cylinder with a cone removed — each cut at the same height, with the disc and the annulus the cut produces marked and their equal areas given.

The slice that has to match

The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.

geometry · Circle area Each doubling adds a quarter of what the last one added. n=6: A=2.5980762114; n=12: A=3.0000000000; n=24: A=3.1058285412; sliver ratio 0.263305.

The error that predicts itself

Archimedes' polygons close on π slowly — six doublings of the sides for two decimals. But the error does not fall at random. Each doubling adds almost exactly a quarter of the area the doubling before it added, so the rest of the slivers can be summed before they are drawn. Liu Hui saw the quarter in 263; Snell and Huygens turned it into bounds; and the 96-gon that gave Archimedes two decimals gives six once its error is known.

geometry · Circle area

Named alongside it

The objects these essays reach for when they reach for this one.

Method of exhaustionCircle areaLimitPiApproximationAreaConvergence rateDissectionGeometric seriesParabola

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